Physics 2220 Fall 2010

George Williams

FINAL EXAM - REVIEW PROBLEMS

A data sheet is provided. Reminder: All previous review sets are fair game. Solutions are available on the course
web site. There is no solution for problems marked with *.

1. (a) A diamond (n = 2.42) is under water (n = 1.33). Calculate the polarizing angle for light incident on
the diamond.
(b) Yellow light of wavelengths 589 nm, is incident normally on a single slit. The second minimum from
the center is 2.75 cm from the center of the pattern on a screen 4.35 m away from the slit. Calculate
the slit width.
(c) Calculate the focal length of the lens shown in air. The lens is made of glass with
n = 1.65.

(d) Completely unpolarized light is incident on three polarizers.

The polarization axes of A and B make an angle of 30.0° with
each other and the axes of B and C make an angle of 55° with
each other. Calculate the intensity that comes out of C as a
function of I o.

(e) Red light of 8 = 635 nm, shines through a fine wire screen. The first maximum from the center
occurs at 2 = 1.00°. The sixth maximum (the center counts as zero) is missing. This is the first one
missing. Calculate the size of the open space between the wires of the screen.

2. An object is placed 30 cm in front of lens A.

(a) Calculate the position of the final image as a distance

along the optic axis from lens B. State clearly whether
the final image is right or left of lens B.
(b) State whether the final image in (a) is erect or inverted
and justify your answer.
(c) If the original object is 1.00 cm high, what is the size of
the final image?
(d) State clearly and give your reason for whether the final
image is real or virtual.
NOTE: Signs are important in this problem and sign errors will not be treated as "trivial." In (a) and (c), no
credit if a reason for your answer is not given.

3. Two optically flat, thick, glass plates with index of refraction n = 1.55, are
set up as shown. At one edge the plates touch, and at the other edge they
are separated by placing a hair between them. At the position of the hair
there is bright interference band in reflection for M = 500 nm (green) and
for 566.7 nm (yellow). There is no other place between the hair and the
touching edge where the green and yellow maxima exactly coincide.

(a) How thick is the hair?

(b) Calculate the next wavelength, longer than 567 nm, that will also show a maximum at the position of
the hair?
4. Two slits of width 8.00 :m have their centers a distance 30.0 :m apart. Blue light ( 8 = 460 nm)is incident
normally on the slits. If the intensity of the center (m = 0) line is taken as I o, calculate the intensity I at
2 = 4.20° as a multiple of Io. [Note: The slits have finite width.]

5. (a) Find the critical angle for total internal reflection for an oil-air interface if n oil = 1.25.
(b) Find the thickness of a quarter wave plate for yellow light ( 8 = 589 nm), if n s = 1.5678 and
n f = 1.5213 for the material of the plate.
(c) A sugar solution has n R = 1.34500 and n L = 1.34400. Find the angle through which the plane of
polarization is rotated if plane polarized light of wavelength 8 = 589 nm passes 10.0 cm through this
solution.
(d) Find the focal length for the lens shown. The lens is in air
(n = 1.55).

(e) The third interference maximum from the center is found at 6.75 cm from the center of the screen in a
two-slit interference pattern. If the light is green ( 8 = 500 nm) and the distance to the screen is
3.75 m, find the slit separation.

6. (a) Find the focal length of the lens shown. It is made of glass of
index n = 1.55.

(b) Light reflected from a diamond (n = 2.42) is observed to be completely polarized. What is the angle
(measured from the normal) of incidence?
(c) Two slits are 1.42 × 10 -6 m apart. If green light is normally incident, how far apart are the
interference maxima for m = 1 and m = 3 (on the same side of center) on a screen 5.00 m away?
(d) W ith a substance of unknown index of refraction on top of a glass plate, the critical angle for total
internal reflection is observed to be 72.0°. The index of the glass if 1.503. W hat is the unknown
index?
(e) Light is incident from the left on the system shown. The
principle axes of the quarter wave plate are at 45° to the plane
of polarization. After being reflected by the mirror and passing
again through the quarter wave plate, what happens to the light
when it passes the polarizer the second time?
_________________________________________________

7. (a) For a single slit illuminated by red light ( 8 = 650 nm) the third minimum on a screen 10.0 m away
from the slit is 3.75 cm from the center. Find the wavelength for which the fifth minimum is the same
distance from the center.
(b) A two-slit pattern shows the 7th, 14th, 21st, etc. maxima are missing. If the slit spacing is 0.0250 cm,
what is the slit width?
(c) It is found the total rotation of the plane of polarization by 10.0 cm of a sugar solution is 127°. W hat
is the difference between n r and nl?
 (d) Three polarizers are arranged with the transmission axes of
each rotated 30° from the one before. W hat is the maximum
intensity of light (as a fraction of the incident intensity I o)
that can pass this system?

(e) Calculate the polarizing angle for a diamond (n = 2.42) submerged in water (n = 1.33). The incident
light is from the water side of the boundary.

8. Given a thin lens of focal length 50 cm. An object 2 cm height

is placed 2 m to the left of the lens.

(a) For the image, find the position, size and whether or not
it is inverted.
(b) A second lens of f = 100 cm is placed at position A. For
the resulting image, find the position, size and whether it
is erect or inverted.
(c) The same second lens is placed at position B. Repeat
the calculations done in (b).
(d) The same second lens is placed at position C. Repeat
the calculations in (b).

9. Given the lens system shown. The focal lengths are fA = -40.0 cm and
fB = +140 cm. The object is 1.75 cm high.

(a) Find the location of the final image with respect to lens A. Take
distances to the right of A as positive, distances to the left of A
as negative.
(b) How high is the final image?
(c) Is the final image erect or inverted with respect to the object?
(d) Is the final image real or virtual?

10. Given the lens system

shown. The object is 60 cm to the left of lens A.

(a) Find the position of the final image measured in cm to

the right or left (state clearly) of lens B.
(b) Characterize the final image as erect or inverted, real or
virtual.
(c) If the original object is 1.75 cm high, what is the height
of the final image?

11. A and B are both diverging lenses. The magnitude of their focal
lengths are A = 50.0 cm and B = 125.0 cm. An object is placed
25.0 cm to the left of A.

(a) Find the position of the final image measured from A,

positive to the right, negative to the left.
(b) If the object is 2.00 cm high, find the size of the final
image.
(c) Is the final image real or virtual? W hy?
(d) Is the final image erect or inverted? W hy?
12. A thin flake of mica (n = 1.6) is used to cover one slit of a double-slit arrangement. The central point on the
screen is now occupied by what used to be the seventh bright fringe. If 8 = 6500 A (650 nm), what is the
thickness of the mica?

13. Plane wavefronts of light are incident normally upon two narrow slits separated by a distance d (slit width <<
d). On a very distant screen an interference is observed with adjacent intensity maxima separated by a
distance of 0.00500 m. The entire experiment is now repeated under water (n = 1.33). Now what is the
spacing of the adjacent maxima?

14. A Newton's rings experiment is performed with oil of index n =

1.65 between the two surfaces and is observed in reflection.
Assume an index of glass of n = 1.50. Green light of
wavelength 520 nm is incident normally. The radius of
curvature of the curved surface is 100.0 cm.

(a) Is the center spot light or dark?

(b) Find the radius of the fifth dark fringe. (If the center is
dark, count it as one.

15. A thin film of oil (refractive index 1.25) is floated on a thick glass plate (refractive index 1.50). Plane light
waves of variable wavelength are incident normal to the film. W hen one views the reflected wave it is noted
that complete destructive interference occurs at 6000 A and constructive interference at 7000 A. Calculate
the thickness of the oil film.

16. A very thin plate of glass (n = 1.540) is placed under water (n = 1.330) and illuminated at perpendicular
incidence with white light. The plate is 1.786 × 10 -6 m thick. Find all of the wavelengths in the visible
region of the spectrum for which there is destructive interference for the light transmitted through the plate.
(Visible light = 400 to 700 nm.)

17. Red light from a laser ( 8 = 650 nm) is incident normally (angle of incidence = 2) on a diffraction grating.
The first maximum on either side of the center spot occurs at an angle of 23°, as in Fig. (a). Now the angle of
incidence is changed to 15°. Calculate 21 and 22 in Fig. (b), the direction of the first maxima on each side of
center. 22 might be positive or negative, so use the sign convention given and measure the angles from the
normal. (This problem involves rederiving the grating equation for angles of incidence other than 0°.

18. In a single slit diffraction experiment the 7th minimum from the center in red light ( 8 = 650 nm) is observed
at 15.7 cm from the center of a screen 3.25 m from the slit.

(a) Find the position of the 8th minimum in green light ( 8 = 500 m).
(b) Find the distance between the 8th and 9th minima on the screen for the green light.

19. In a single slit diffraction pattern the distance between the first minimum on the right and the first minimum
on the left is 5.2 mm. The screen on which the patten is displayed is 80 cm from the slit and the wavelength
is 5460 A (5460 A = 546 nm). Calculate the slit width.
20. A single slit diffraction pattern is observed in air using red light of 8 = 650 nm. The first minima on either
side of the center maximum are observed on a screen 6.50 m from the slit to be 1.25 cm apart.

(a) Calculate the slit width.

(b) Calculate the separation of the same two minima if the experiment is repeated under water (n = 1.33)
with the slit width and the screen distance the same.

21. (a) W hat is the critical angle for total internal reflection of a diamond immersed in water (use light of
8 = 589 nm).
(b) If the microwaves in your oven have a frequency of 800 MHZ, calculate their wavelength.
(c) A diffraction grating with 26,500 lines/inch is used with yellow light of wavelength 589 nm in a
perpendicular incidence. W hat is the largest possible order that can be observed in the interference
pattern?
(d) Sunlight is incident on a solar reflector with an intensity of 655 W /m 2. Calculate the peak value of the
electric field in this light beam.
(e) Calculate the force on 1.00 m 2 of the solar collector in (d) if 20% of the light is reflected and 80
absorbed. (Intensity is the same as in (d).)

22. Lens A has a focal length whose magnitude is 55.0 cm. Lens B
has a focal length whose magnitude is 65.0 cm. (You supply the
signs.) An object is placed 98.0 cm in front of lens A. The
system is in air.

(a) Calculate the position of the final image of the system

expressed as a distance to the right or left of lens B.
(b) Is the final image erect or inverted?
(c) If the original object is 1.00 cm high, how high is the
final image?

23. In a single slit diffraction the seventh minimum from the center of red light ( 8 655 nm) is at the same position
on the screen as one of the minima for violet light ( 8 = 417 nm). The screen is 5.50 m from the slit and the
position of these two minima is 3.75 cm from the center of the pattern.

(a) W hich minima for violet light is at the position given?

(b) W hat is the width of the slit?

24. A soap film, whose index of refraction is 1.34, is observed in air in transmission with light perpendicular to
the film. Constructive interference in observed for the following wavelengths, among others. There is no
guarantee that these are consecutive. Calculate the minimum thickness of the film consistent with the data
given.

81 = 655.0 nm (red) 82 = 467.8 nm (blue)

83 = 409.4 nm (violet)

25. It is desired to measure the thickness of a thin flake of mica (n = 1.60). W hite light is incident perpendicular
to the surface. Complete destructive interference in reflection is observed for the following wavelengths.
Find the thickness of the mica. (Note: Round-off errors can be confusing in this problem.)

81 = 611 nm (red-orange) 82 = 550 nm (yellow)

83 = 500 nm (green) 84 = 459 nm (blue)
85 = 423 nm (violet)
26. (a) Calculate the polarizing angle for light reflected from glass (n = 1.55) under water (n = 1.33).
(b) There are 4 polarizing screen in a row. The polarizing
direction of each is rotated 25.0° from the one before.
Calculate the intensity, after D, as a fraction of I o.

(c) In a two-slit experiment, the 5th minimum from the center of the pattern is observed at 3.75 cm from
the center on a screen 5.25 m away from the slits, using light of 8 = 590 nm. Calculate the slit
separation. The minimum described is not the single slit minimum, but is a minimum in the two-slit
pattern.
(d) Calculate the thickness of a quarter-wave plate for yellow light of wavelength 590 nm, if n fast = 1.7500
and n slow = 1.7520 for the material of the plate.
(e) In an optical rotation experiment the plane of polarization for a wavelength of 590 nm is rotated 235°
in a cell 10.00 cm long. Calculate the quantity (n r - n R).

27. An object is set up 60.0 cm to the left of lens A. The other

dimensions and focal lengths are as shown.

(a) Calculate the position of the final image as a distance

along the optic axis measured from lens A. State clearly
whether the image is to the right or left of lens A.
(b) Calculate the magnification of the system.
(c) State clearly and give a reason for whether the final
image is erect or inverted.

28. In a grating experiment, light is perpendicularly incident on the grating.

Green light of wavelength 505 nm, is found to give a third order
maximum at 2 = 35.00°. A different wavelength is found to give a
maximum in second order at 2 = 22.00°.

(a) Find this second wavelength.

(b) If we assume extremely narrow slits, calculate the intensity for
505 nm at 2 = 32.0°, expressed as a fraction of the intensity in the
exact center of the pattern.

29. A film of oil is set up on the surface of the water as shown.

Light is incident perpendicular to the surface from the air side.
The film is 1.45 × 10 -6 m thick.

(a) Calculate all of the wavelengths in visible light (400-700

nm) that show constructive interference maxima in
reflection.
(b) Calculate all of the wavelengths of visible light that
show constructive interference maxima in transmission.
30. (a) W hat is the maximum order that can be observed with a grating with 1550 lines/cm, using light with
wavelength 475 nm.
(b) Calculate the polarizing angle for light of wavelength 650 nm for light in water (n = 1.33) incident on
a diamond (n = 2.40).
(c) Calculate the focal length in water (n = 1.33) for the glass (n = 1.55) lens
shown. You supply the signs for ,R ,.

(d) Calculate the thickness of a quarter-wave plate for a material where n f = 1.4700 and n s = 1.4750 for
8 = 590 nm.
(e) Five polarizers are arranged in a row. Completely
unpolarized light is incident on the first. The angles between
their axes are 10° each. W hat is the intensity of the emerging
light as a fraction of I o?

31. Two lenses are arranged as shown. You supply the sign for the
focal length. The object is 3500 mm to the left of A.

(a) Find the position of the final image measured from the
position of B. State clearly whether it is right or left of
B.
(b) W hat is the magnification of the system?
(c) Is the image erect or inverted? Show clearly why.
(d) Is the image real or virtual? State clearly why.

32. A two slit interference experiment is performed with two colors of light. The seventh order (m = 7, where the
center is m= 0) fringe for light of 8 = 555 nm is at 9.50 cm from the center on a screen 5.00 m from the two
slits.

(a) Calculate the wavelength of light that will have the fifth order fringe at 9.50 cm from the center.
(b) Calculate the slit separation.

33. A regular hexagon is constructed of glass with an index of refraction of

1.55. A beam of light is incoming in the direction shown. There are two,
and two only, paths for this light striking at different points on face A to
go through the hexagon. Find the face from which each beam leaves the
hexagon and its direction with respect to the normal to that face.
34. (a) W hat is the critical angle for total internal reflection for light from diamond (n = 2.42) to water
(n = 1.33).
(b) Calculate the focal length of the lens shown. The lens is made of
plastic with an index of n= 1.40.

(c) If the speed of red light is 1.93 × 10 8 m/s, and blue light is
1.89 × 10 8 m/s in a sample of glass, calculate the angle between
the two colors after they enter a glass plate at 45° to the normal.
____________________________________________________

(d) In a 2-slit pattern the position of maxima on a screen are at y = 0, ± 2.00 cm, ± 4.00 cm, etc. If the
intensity is I o at y = 0, what is the intensity at y = 1.67 cm?
(e) A soap film (n = 1.33) is 800 nm thick and surrounded by air. Calculate the longest wavelength of
light for which there will be constructive interference in reflection at normal incidence.

35. Light is incident parallel to the base of a 30° !60° !90° prism made of glass
with n = 1.520. It enters close enough to the base that it strikes face C after
refracting at the first surface. W hich face of the prism does the main beam
emerge from? Calculate the angle of this beam with respect to the normal to
that face.

36. For the lens system shown, the object is 150 cm to the left of lens A.

(a) Find the position of the final image, as a distance to the right or
left (specify) of LENS A.
(b) Calculate the magnification, including sign.
(c) Is the image real or virtual?
(d) Is the image erect or inverted?

37. A film of oil is placed on water (n = 1.33). The film is viewed in

reflection from above, with the light at normal incidence. W hen the
film is extremely thin, it reflects white light. W hen the film is
1050 nm thick, interference maxima are observed for light of
wavelength 800 nm, 480 nm and 300 nm, among others. (No
guarantee that these are consecutive.) W hat index of refraction do
you infer for the oil from these data?
38. (a) Calculate the polarizing angle for light reflected from glass (n = 1.55), if the glass is immersed in
carbon tetrachloride (a liquid with n = 1.46).
(b) Calculate the thickness of a quarter-wave plate at 8 = 550 nm, if the two indices of refraction are n =
1.4250 and n = 1.4355.
(c) Linearly polarized yellow light ( 8 = 589 nm) passes through 1.00 cm of a sugar solution. The plane
of polarization is found to be rotated by 165°. If the average index of refraction is 1.3500, calculate
the indices for right and left circularly polarized light (which is which cannot be determined from this
information). Note: Many more than three significant figures are needed here.
(d) Calculate the critical angle for light in glass (n = 1.5500) approaching a surface covered with benzene
(n = 1.5014).
(e) Calculate the focal length in air of the lens shown. Take the index of the lens material as n = 1.440.

39. A prism is constructed of diamond (n = 2.40) with an apex angle of

25°. A pencil beam of light is incident to the base of the prism. Outside
is air.

(a) Show which face, A or B, the light emerges from.

(b) Calculate the angle of the emergent beam from the normal to that
face.
(c) Show on a clear drawing the direction of the emergent beam
(which side of the normal it is on).

40. For the lens system shown in air the original object is 250 cm to the
left of lens A. Focal lengths are given as a magnitude only, you
supply the sign.

A: *f* = 100 cm
B: *f* = 50.0 cm
C: *f* = 150 cm

(a) Calculate the position of the final image as a distance left ( !) or right (+) of lens C.
(b) Is the final image real or virtual. Give your reason.
(c) Is the final image erect or inverted?
(d) Calculate the magnification of this system.

41. A thin film of oil (n = 1.30) is sandwiched between two thick glass plates (n = 1.55). If constructive
interference is observed in transm ission with light at perpendicular incidence at the following wavelengths
(not necessarily consecutive), calculate the thickness of the film.

81 = 550 nm
82 = 770 nm
83 = 427 nm
84 = 350 nm

42. Given a spherical nonconductor where the volume charge distribution is

given by D = Doe !8R, and D at R = R o is Do/2. [ D = 0 for R > R o]

(a) Find an expression for 8.

(b) Calculate the total charge, Q o, on the sphere.
(c) Find the electric field at point P 2 at radius R 2, outside the sphere.
(d) W hat is the electric field at point P 1 at radius R 1, inside the
sphere?
 43. Given the circuit shown with the numerical values given.

(a) Calculate the value of the current in R 4.

(b) Find the power being dissipated in R 2.
(c) Determine the total power being delivered by the battery to the circuit.

g = 275 V; R 1 = 375 S; R 2 = 100 S; R 3 = 80.0 S; R 4 = 120 S; R 5 = 200 S

44. * Given the network shown and values given.

(a) Calculate the charge and potential on C 3 and C 4 when the switch is
closed.
(b) The switch is closed for a long time and then opened and left open.
A dielectric with 6 = 3.00 is inserted in C 4. Calculate the potential
and charge on all capacitors.

g = 150 V; C 1 = 3.00 :F; C 2 = 4.00 :F;

C 3 = 2.00 :F; C 4 = 1.00 :F

45. * Use the Biot-Savart law to calculate the magnetic field at point P (in x^ , y^ , z^
notation), due to a current of 5.00 amperes in the direction shown by the
arrows. The round portion of the wire is circular (R = 6.00 cm), and P is at the
center. For ease in grading, label the infinite straight segments (1) and (2) as
shown in the figure.

46 Given the circuit shown.

(a) Calculate the charge on the capacitor 1.50 time constants after the
switch is closed. (Numerical value.)
(b) Find the current in R 3 at t = 4. (Numerical value.)
(c) Calculate the charge on the capacitor 0.350 s after the switch is opened,
after being closed for a long time.
(d) Calculate, in complete detail, the time constant for charging the
capacitor. No short cuts from other classes allowed. (Numerical value.)

g = 90.0 V; R 1 = 150 S; R 2 = 200 S; R 3 = 300 S; C = 450 :F

47. If the electric field at the Earth's surface is +170 V/m, and 1000 m above the surface is +112 V/m, calculate
the energy stored in the electric field in a cube 1000 m on a side with the bottom at the Earth's surface.
Assume the variation in E is linear in the distance above the surface. (That is, use E = E o + ah, and evaluate
a.)

48. Initially all capacitors are discharged.

(a) If the switch is closed at t = 0, calculate the current in R as a function of

time, evaluating all numerical constants.
(b) After being closed a long time, the switch is opened, and then a
dielectric with 6 = 4.25 is inserted in C 2. Calculate the charge and
potential on each capacitor after this event.
(c) After (b) the switch is closed again. How much additional charge flows
from the battery to the capacitors if you wait a long time?

g = 125 V; C 1 = 650 pF; C 2 = 275 pF; C 3 = 400 pF; R = 1.25 × 10 5 Ohms

49. * A long, hollow, cylindrical copper pipe, with outer radius R o = 2.75 cm and
inner radius R i = 0.50 cm, carries a current of I = 1.46 × 10 4 A. The current is
uniformly distributed.

(a) Calculate the magnitude of the magnetic field at r = 0.45 cm from the
center of the cylinder.
(b) Calculate the magnitude of the magnetic field at r = 3.75 cm.
(c) Calculate the magnitude of the magnetic field at r = 1.25 cm.

50. For the circuit shown the switch is open for a long time.

(a) If the switch is closed at t = 0, calculate the charge on the capacitor

after 1.00 time constants have elapsed. (Numerical value. Full loops
and junctions not required.)
(b) If the switch is closed for a long time, and opened at t = 0, calculate
the time constant for discharging the capacitor. (Full loops and
junctions not required.)
(c) If the switch is closed for a long time, and opened at t = 0, calculate
the charge on the capacitor when t = 2.00 time constants. (Numerical value.)

g = 275 V; R 1 = 15,000 S; R 2 = 12,500 S; R 3 = 6,750 S; C = 4.20 × 10 !6 F

51. Given a long, cylindrical piece of non-conductor, of radius R o. The charge

distribution in the non-conductor is given by D = Ae !6r and D = 0 for r > R o.
A and 6 are positive constants.

(a) Calculate the magnitude of the electric field at point P, a distance r

from the center of the cylinder.
(b) Calculate the magnitude of the electric field at an arbitrary point Q
outside of the cylinder, a distance r from the center, where r > R o.
(c) Calculate the energy stored in the electric field between r = R o and
r = 2R o for a 1.00 m length of the cylinder.

52. Given a rod of nonconductor of length L with a charge density expressed

as 8 = 8ox 2 C/m.

(a) Calculate the x-component of the electric field due to the rod at
point P, a distance a from the end of the rod.
(b) Calculate the electric potential at point P due to the rod.

53. (a) If the earth's magnetic field in Salt Lake City is 0.750 gauss downwards at an angle of 75.0° from
horizontal, calculate the magnetic flux, in T Am 2, through a rectangular section of the physics parking
lot that is 6.00 × 9.00 meters.
(b) Calculate the torque on a coil of wire of 17 turns, radius 2.50 cm, carrying
a current of 1.25 A. The plane of the coil is at an angle 2 = 27.0° with
respect to the magnetic field, as shown. B = 375 gauss.
 (c) Two long, parallel wires carry the currents shown. Calculate the force,
magnitude AND direction on 3.25 m of the upper wire.

(d) For the expression 1/(x 2 - a 2) 7/2 use the binomial expansion and calculate completely the third term
(the term in a4). (Assume x << a.)
(e) If all capacitors have the same value, C, calculate the effective capacitance
between a and b.

(f) Calculate the electric potential, magnitude and sign at the point midway
between charges A and B. A = 3.00 × 10 -9 C; B = - 4.25 × 10 -9C.

54. Given the circuit shown.

(a) Find the current in R 5.

(b) Calculate the potential across R 2.
(c) Determine the power being dissipated in R 4.

g = 175 V; R 4 = 120 ohms; R 1 = 450 ohms; R 5 = 270 ohms;

R 2 = 85 ohms; R 6 = 350 ohms; R 3 = 65 ohms; R 7 = 150 ohms

55. Initially the switch in the circuit is closed for a long time. Initially there is
a dielectric with 6 = 4.25 in C 3. (All capacitor values are given without
dielectric.) Now the switch is opened.

(a) Calculate the charge and potential on each capacitor.

(b) After (a) the dielectric is moved from C 3 to C 1 with the switch still
open. Calculate the charge and potential for each capacitor in this
new situation.

g = 250 V; C 1 = 90 :F; C 2 = 125 :F; C 3 = 40 :F

56. Assume a spherically symmetric charge distribution on a non-conducting

sphere. The charge density is given by D = BR 3, for R < R o and D = 0 for R >
R o. B is a constant.

(a) If the total charge is Q, calculate B in terms of Q, R o and numbers.

(b) Calculate the electric field a distance R o above the surface of the
sphere.
(c) Determine the electric potential at R = R o.
(d) Find the electric potential difference between R = R o/2 and R = R o.
Express this as V(R o) - V(R o/2), and get the sign right if the charge distribution is of negative charge.
57. For the circuit shown the switch is open for a long time and then closed for
exactly 2.00 time constants, and then opened.

(a) Calculate the charge on the capacitor the instant the switch is
opened.
(b) Find the magnitude of the current in R 3 6.00 × 10 -3 seconds after the
switch is opened.
(c) As discussed in class, show in detail how to obtain the time constant
for charging the capacitor and obtain a numerical value for that time
constant.

C = 2.40 × 10 -6 F; g = 225 V; R 1 = 1.50 × 10 3 ohms; R 2 = 0.700 × 10 3 ohms; R 3 = 1.10 × 10 3 ohms

58. Consider a spherical uniform charge distribution of radius R o.

(a) Find the value of r such that half the total charge is inside r and call this value r 1.
(b) Find the electric field at r = r1.
(c) Calculate the energy stored between r = ½ r 1 and r 1.

59. Consider a piece of nonconductor in the shape of a coin with a hole in the center. The
material has a charge density given by D = Dor between r = r 1 and r = r 2 and zero
everywhere else. D is charge per unit area in C/m 2.

(a) Calculate the electric potential at the center.

(b) Find the electric field at the center.

60. (a) Calculate the cyclotron frequency, in Hz, of an electron in a magnetic field
of 3330 gauss.
(b) Calculate the capacitance of a parallel plate capacitor which is circular with a radius of 1.57 m and
a plate separation of 1.10 mm. There is no dielectric.
(c) Calculate the magnetic dipole moment of a rectangular coil of wire consisting of 327 turns ,
carrying a current of 2.34 A, with a length of 4.26 and a width of 2.25 cm.
(d) A 12.0 pF capacitor is charged to 110 V. It is connected at both ends to an uncharged 17.0 pF
capacitor. Find the potential across the pair of capacitors.
(e) In the circuit shown, the switch is closed for 1.75 s and opened at
t = 0. Calculate the charge on the capacitor at t = 2.25 s.

R 1 = 2250 S
R 2 = 1550 S
C = 3.50 × 10 !3 F
g = 150 V.

(f) Calculate the magnetic energy stored in a toroid with 975 turns carrying 1.75 A. The
cross section of the toroid is square with sides a = 1.80 cm and an inner radius of a/2.

61. A parallel plate capacitor has a plate area of 900 cm 2 and a plate separation of 0.500 cm. The space
between the plates is empty.

(a) W hat is the numerical value of the capacitance?

(b) W hat is the potential difference if the charge on each plate is 6.00 × 10 !8 C?
(c) W hat is the electric field between the plates?
(d) W hat is the energy density between the plates?
(e) W hat is the total energy stored?
62. A metal rod is placed inside a metal cylinder shown in the cross section. Both
are very long. The inner rod is given a charge of +39.2 × 10 !6 C.m The outer
cylinder is given a charge of !67.2 × 10 !6 C/m. The inner rod has a radius of
0.327 cm, and the outer cylinder, which is thin, has radius of 3.75 cm.

(a) Calculate the electric field, magnitude and direction, at r 1 = 0.625 cm.
(b) Calculate the electric field, magnitude and direction, at r 2 = 4.25 cm.

63. Consider an infinite line of charge with a charge density given by !8 C/m. Calculate the work
needed to move a charge +Q from point B to point A. Give both the magnitude and the sign.

64. Consider a positively charged sphere whose charge density can be expressed as D = Do(1 - "R), where the
radius of the sphere is R o. If the charge density goes to zero at R = R o, and Q is the total charge, calculate:

(a) the value of the " in terms of R o;

(b) the value of Do in terms of Q, R o and numbers only;
(c) the electric field at R = R o/3 (eliminate the Do and " from your final expression).

65. (a) In the drawing are shown, three long, straight wires are perpendicular to
the paper. If they each have a current of 5.00 A out of the paper and a is
2.00 cm, calculate the magnitude of the force per unit length on wire A.
________________________________________

(b) If g = 150 V, and all the resistors are 375 S, calculate the current
through R 2 .__________________________________________

(c) If the switch in the drawing is closed for 25.0 s and then opened,
calculate the current in R 2 35.0 s after it is opened. R 1 = 150,000
S, R 2 = 250,000 S, C = 375 :F, g = 200 V.

(d) Calculate the magnitude of the magnetic field at point P due to a

single loop of wire of radius R = 4.75 cm, carrying a current of 375 A. P is
a distance R = 4.75 cm from the plane of the loop along a line
perpendicular to the loop and through its center.

(e) An electron moving in a plane parallel to the earth's surface moves in a circle of radius 0.0736 m.
If the earth's field is 4.00 × 10 -5 T, at an angle of 90.0 / from the horizontal at this point on the
earth, calculate the velocity of the electron.
 (f) The combination of C 1 and C 2 is charged to 125 volts with a
battery. The battery is now removed, and dielectric of 6 = 4.30 is
inserted in C 2 . Calculate the new potential difference.
C 1 = 4.75 pF, C 2 = 1.25 pF.

66. (a) Unpolarized light goes through three polarizing sheets in succession. Sheet 1 is
oriented vertically. Sheet 2 is oriented 30° from the vertical and sheet 3 is 90°
from the vertical. If light of intensity S o is incident on the first, calculate the
intensity of light leaving sheet 3.

(b) Calculate the time constant for the current in the circuit after the switch
is closed.

g = 150 V; R = 650 S; L = 42.0 mH

(c) A diamond (n = 2.40) is immersed in water (n = 1.33). Determine the polarizing angle for
polarizing light in water reflected off the diamond.
(d) For the system in (c), calculate the critical angle for total internal reflection for light emerging from
the diamond into the water.
(e) Calculate the focal length, with sign, for the lens shown. The lens
is made of glass with n = 1.55.

(f) If the earth's magnetic field is 1.20 × 10 !4 T, oriented at 65° to the horizontal, find the
magnetic flux through an area 50 m wide by 100 m long.

67. For the network shown:

(a) Calculate the effective resistance between A and B.

(b) Find the current in R 7.
(c) Find the current in R 4.

g = +150 V R4 = 350 S
R 1 = 75.0 S R5 = 450 S
R 2 = 160 S R6 = 500 S
R 3 = 200 S R7 = 400 S

68. For the circuit shown the switch is open for a long time, and
closed at t = 0.

(a) Calculate the current in R 3 at t = 0.

(b) Calculate the current in R 2 at t = 4.
(c) Calculate the current in R 2 at t = 2 J (two time constants).
(d) If the switch is closed for a long time and then opened at t = 0 (a
new t =0), find a complete expression with all constants evaluated
for the current in R 2.
R 1 = 250 S; R 2 = 550 S; R 3 = 450 S; g = 135 V; L = 4.20 mH
69. A two slit interference experiment is performed with green light of wavelength 550 nm. The centers of the
slits are 0.300 mm apart.

(a) On a screen 4.0 m away from the slits, the interference pattern is observed. W hat is the spacing
between the m = 1 and m = 0 bright bands?
(b) The experiment is repeated with orange-red light. The m = 11 fringe of the red light is in exactly
the same position as the m = 13 fringe of the green light was. Calculate the wavelength of the red-
orange light.

70. A very long copper cylinder carries an electric current of I o Amperes. The radius of the rod is R o and the
current density is uniform.

(a) Calculate the current density in the rod.

(b) Find the magnitude of the magnetic field at R o/3.
(c) Determine the energy stored in the magnetic field between R o and 3R o per unit length.

*71. For the lens system shown, in air, the original object is 190 cm in front of lens A. The magnitude of the
focal length is given. You must supply the sign.

(a) Calculate the position of the final image as a distance right of left
(specify) of lens B. Show all intermediate steps so that appropriate
partial credit can be given.
(b) Is the final image real or virtual? You must give a reason for credit.
(c) Find the magnification of the system.

|fA| = 120 cm
|fB| = 80 cm

72. A long non-conducting rod of radius R o, is electrically charged. The charge density within the rod is
uniform and can be represented as + 8(C/m) along the rod. (A length R of the rod has a total charge 8R.)

(a) Find an expression for the electric field for any value of R > R o.
(b) Find an expression for the electric field for any value of R < R o.
(c) W hat is the magnitude of the potential difference between R o/2 and R o.
(d) Determine the energy stored in the electric field, per meter length, between R o and 2R o.

73. (a) A two-slit interference pattern is projected on a screen 3.50 cm from the slits using a laser light of
wavelength 525 nm. If the slits are 0.200 mm apart and are very narrow, find the distance between
the m = 0 and m = 1 maxima.
(b) A single slit of width 0.050 mm is illuminated with light of 525 nm as above. The distance from
slit to screen is 3.50 m. Calculate the distance on the screen between the m = +1 and m = !1
minima.
(c) W hite light is incident perpendicular on a soap film (n = 1.34) of thickness 0.750 × 10 !6 m.
Calculate ALL the wavelengths in the visible (400 nm to 700 nm) that show constructive
interference in reflection.
(d) Determine the critical angle for total internal reflection for diamond (n = 2.40) immersed in water
(n = 1.34).
(e) Find the polarizing angle for light incident on a diamond under water.
(f) For a diffraction grating with 75.0
lines/mm, calculate the distance from the
center for green light ( 8 = 500 nm) in 3rd
order.
74. A long cylinder shaped charge distribution of radius R o has a charge density given by D = BR 3 (not
physically sensible, but it keeps the math simple). B is a constant.

(a) Find the total charge for a length R of this cylinder.

(b) Calculate the electric field at an arbitrary value of R where R < R o.
(c) Calculate the magnitude of the electric potential difference between R = 0 and R = b R o.

75. For the circuit shown all capacitors are initially uncharged. W ith switch B open,
switch A is closed for a long time and then opened. Now B is closed.

(a) Calculate the potential and charge on each capacitor.

(b) Now a dielectric with 6 = 4.00 is inserted into C 2 (A remains open, B
remains closed). Calculate the new value of the charge and potential
for each capacitor.

g = 150 V; C 1 = 200 pF; C 2 = 300 pF; C 3 = 100 pF, R = 150 S

76. In the circuit shown the switch is closed at t = 0 after being open for a long time.

(a) At t = 0, what is the current in R 2?

(b) Calculate the current in R 3 at t = 4.
(c) Find the current in the inductance at t = 1.2 time constants.
(d) Calculate completely, using loops and junctions as done in class (no short
cuts), the time constant for the current in the inductance with the switch
closed and give a numerical value for this quantity.

g = 175 V; R 1 = 200 S; R 2 = 350 S; R 3 = 250 S; L = 7.80 :H

77. The wire shown has a current I given by I = I o cos Tt.

(a) At t = 0, calculate the flux through the rectangle

shown. The wire and the rectangle are both in the
plane of the paper.
(b) Determine the magnitude of the voltage across the
resistor R as a function of time.
(c) At a time given by Tt = B/2 is the current in the loop
clockwise or counterclockwise? Clearly state your
reason (no credit will be given for guessing).

78. A bolt of lightning has a current density distribution given by j = AR 4 for R < R o and 0 for R > R o (not
physical, but keeps the math simple).

(a) Calculate the magnetic field at a radius R, where R < R o.

(b) Find the energy in the magnetic field for a length R and between R = 0 and R = ¾ R o.

79. For the lens system shown the magnitudes of the focal lengths are
given. You supply the signs. The original object is 800 cm to the
left of lens A. The lenses are 200 cm apart.

(a) If lens B was not there, calculate the position of the image
as a distance right or left of lens A.
(b) For (a) state the nature of the image (real, virtual, erect,
inverted).
(c) W ith lens B in place, calculate the position of the final
image as a distance right or left of lens B.
(d) State the nature of the final image (real, virtual, and erect or inverted with respect to the original
object).
80. Three long parallel wires, shown in cross section, carry currents as shown. +
means out of the paper; - means into the paper. The wires are at three corners
of a square of side a (a = 4.35 cm).

(a) Calculate the magnetic field at C due to wires A and B. Give

magnitude and direction where the direction is an angle measured from
the positive x-axis. Positive angles are counterclockwise.
(b) Calculate the force, magnitude and direction, on each meter of wire C.
Express direction as above.

81. For the circuit shown the switch B is open. The switch A is closed for a
long time and then opened.

(a) Calculate the charge and potential on capacitors C 1 and C 2.

(b) Now, switch B is closed, with A still open. Calculate the
potential and charge on each capacitor.
(c) W ith A remaining open and B remaining closed, a dielectric
with 6 = 3.5 is inserted in C 2. Calculate the new values of
potential and charge on each capacitor.

g = 140 V; R = 350 S; C 1 = 170 pF; C 2 = 200 pF; C 3 = 150 pF

82. A long cylindrical non-conductor has the cross-section shown. Charge is

confined to the shaded region and is governed by the charge density given
by D(R) = DoR 3. D(R) = 0 for R > b or R < a. The charge is positive.

(a) Calculate the total charge in a length R of this cylinder.

(b) Calculate the electric field at a point R where R < a.
(c) Calculate the electric field at a point R where R > b.
(d) Calculate the electric field at a point R where a < R < b.

83. (a) Calculate the energy, in Joules, of a photon of light of wavelength 650 nm.
(b) Calculate the critical angle for total internal reflection for glass (n = 1.65) in air.
(c) In a two slit interference pattern, the maxima on the screen are 1.25 cm apart. If the screen is 4.00
m from the slits, and the slit separation is 2.00 × 10 - 6 m , what is the wavelength of the light?

(d) A soap (n = 1.33) film is observed to reflect light of perpendicular incidence in the yellow (600
nm) and the green (500 nm), and no color in between. W hat is the minimum thickness of the soap
film?
(e) Calculate the focal length of the lens shown in air. The lens is made of
glass with n = 1.55.

(f) An electron is accelerated through a potential difference of 10,000 volts. Calculate its DeBroglie
wavelength.
84. For the circuit given, all switches are initially open. S 1 is
closed for a long time, and then opened. W ith S 1 and S 3 open,
S 2 is closed.

(a) Calculate the potential across C 1.

(b) Find the potential across C 3.

W ith S 2 still closed and S 1 open, close S 3 at t = 0.

(c) Calculate the charge on C 1 3.00 s after S 3 is closed.

C 1 = 1.00 :F; R 1 = 100 S; C 2 = 2.00 :F; R 2 = 1.75 × 10 6 S; C 3 = 3.00 :F; g = 165 V

85. For the lens system shown, the object is 90 cm to the left of the first lens, and 2.00 mm high. You supply
the signs of f.

(a) Determine the position of the final image of the system. Express this in terms of the x-axis scale
given.
(b) Calculate the height of the final image.
(c) Is the final image real or virtual. For credit, you must give a reason.
(d) Is the final image erect or inverted with respect to the initial object. For credit, you must give a
reason.

86. A toroid with rectangular cross section as shown has 1325

evenly spaced turns of wire. The wire carries a current of
1.75 A.

(a) Calculate the magnetic field inside the toroid at

an arbitrary value of r for a < r < b.
(b) Calculate the self-inductance of the toroid.
(c) Calculate the magnetic energy stored between
r = 3.00 cm and r = 3.50 cm in the toroid. [This
Top View Cross Section
is shown as the cross-hatched region.]

a = 3.00 cm; b = 4.25 cm; h = 1.10 cm

87. A spherical distribution of negative charge on a non-conductor is described by a

charge density D = AR 4, where A is a constant. The radius of the sphere is R o

(a) If the magnitude of the total charge on the sphere is Q o, calculate the
magnitude of A.
(b) Calculate the magnitude of the electric field at an arbitrary interior point
(R < R o) of the sphere at a distance R from the center.
(c) Calculate the magnitude of the potential difference between R = R o/2
and R = R o.
(d) Determine the sign of V(R o) - V(R o/2). For credit, you must clearly
explain your reasoning.
88. Shown is the circular cross section of a long copper rod of diameter 1.00 cm.
The current density inside the rod can be expressed as

Here R o is the outer radius of the rod.

(a) If J o = 75 Amps, calculate the total current, including the direction, in

the wire.
(b) Calculate the magnitude of the magnetic field at point A, a distance a = 2.00 cm from the center of
the rod.
(c) Calculate the magnitude of the magnetic field at point B, where b = 0.25 cm from the center of the
rod.

89. The long wire at the top has a steady current of magnitude I o in
the direction shown.

(a) Calculate the magnitude of the magnetic flux for the

rectangle below the wire. Both the rectangle and the
wire are in the plane of the paper.
(b) If the current in the wire decays according to I = I o e - Dt.
Calculate a general expression for the current in the
rectangle as a function of time. The rectangular loop has
resistance = R.
(c) For full credit, clearly explain the direction of the current in the rectangle, clockwise or counter-
clockwise.

90. (a) For the arrangement of charges shown, calculate V B - V A.

(b) In a single-slit diffraction pattern, the second minima (from the center) is found at 3.25 cm on a
screen 4.75 m from the slit. The light is green with wavelength 510 nm. W hat is the slit width?

(c) Calculate the capacitance of a metal sphere whose diameter is 9.45 cm.
(d) In a two-slit interference experiment the third maxima from the center (counting the center as zero)
for red light of wavelength 660 nm is in exactly the same position on the screen as the fourth
maxima for a second color of light. Calculate the wavelength for this second color.
(e) Calculate the energy, in joules, of photons of light with wavelength 550 nm.
(f) A velocity selector is built with an electric field of 11,500 V/m. Calculate the B field needed to
select a velocity for electrons of 2.l0 × 10 6 m/s.

91. Three long, straight wires are perpendicular to the paper with the currents
shown. The wires are at the corners of an equilateral triangle of side a.

(a) Calculate the magnitude of the magnetic field at point P.

(b) Calculate the direction of the magnetic field at P. Show with a
drawing how you define this direction (clockwise or counter
clockwise from a given axis).
 92. For the circuit shown, all resistors have the same value R.

(a) Calculate the total current provided by the battery.

(b) Calculate the current in resistor A.
(c) Calculate the current in resistor B.

93. In the circuit shown the switch S is closed at t = 0 after being open for a
long time.

(a) Calculate the voltage across R 3 at t = 0.

(b) Calculate the voltage across R 1 at t = 4.
(c) Calculate the time constant for charging the capacitor. Show
all steps. No short cuts from other classes.
(d) The capacitor is fully charged and then the switch is opened.
Calculate the charge on the capacitor after 0.030 seconds have
elapsed.

g = 175 V; R 1 = 150 S; R 2 = 200 S; R 3 = 275 S; C = 45.0 :F

94. A conducting rod of mass M is moved to the right with a constant

force. The frictionless rails are horizontal and conducting. The
rod starts with V = 0 at t = 0. The rails are uniformly long. B is
uniform, perpendicular, and into the page. Express all answers
in terms of F, M , R, R, and B, as needed.

(a) Calculate the acceleration of the rod at t = 0.

(b) Calculate the acceleration of the rod at t = 4.
(c) Calculate the current in the rod at t = 4.
(d) W rite an expression for the velocity of the rod as a function of time.

95. Given a spherically symmetric charge distribution, where the charge density is given by:

D(R) = B(1 - "R 4 )

(a) At R = R o the charge density is D(R o) = 0. Calculate ".
(b) If the total charge is Q o, calculate B.
(c) Calculate the electric field at an arbitrary value of R, for R < R o (inside the
sphere).
(d) Calculate the magnitude of the potential difference between the center and the outer surface of the
sphere.
(e) If V = 0 at R = 4, and the sphere is negatively charged, what is the potential at R = 0?

96. W e model the current distribution in a cylindrical bolt of lightening where the current
density is given by:

j(R) = jo (1 - "R 2 )

where R o = 10.5 cm.

(a) If j(R o) = 0, find the value of ".

(b) If the total current is 10,000 A, calculate j o.
(c) Calculate the magnitude of the magnetic field at an arbitrary value of R, where R < R o.
(d) Calculate the magnetic energy stored in a 3.00 m length of this cylindrical bolt.
97. (a) For the arrangement of charges shown, calculate V B - V A.

(b) Calculate the capacitance of a metal sphere whose

diameter is 17.40 cm.
(c) Photoelectrons of energy 1.15 eV are emitted from a metal
surface when light of 8 = 250 nm (ultraviolet) strikes the surface. Calculate the work function of
that metal (in either eV or joules).
(d) In a 2-slit interference experiment red light ( 8 = 600 nm) is used. On a screen 5.00 m from the
slits the fifth maximum (counting the center as zero) is found 7.35 cm from the center. Calculate
the slit separation.
(e) Calculate the polarizing angle for light on a glass (n = 1.57) surface that is
under water (n = 1.33). Light is coming from the water side.

(f) Calculate the critical angle for total internal reflection for a diamond (n =
2.42) immersed in an oil (n = 1.72).

98. For the circuit shown;

(a) Calculate the current in R 1.

(b) Calculate the current in R 2.
(c) Calculate the current in R 3.

g1 = 175 V; g2 = 125 V; R 1 = 3000 S;

R 2 = 2000 S R 3 = 4000 S; R 4 = 5000 S

99. The wire and the rectangular loop shown are in the plane of the
paper.

(a) If the wire has a steady current I, find the flux through the
rectangular loop.
(b) If the current in the wire is given by I = I o cos Tt,
calculate the current as a function of time in the
rectangular loop. The total resistance of the loop is R.

100. A spherically symmetric charge density is given by

for R < R o, and zero everywhere else. C is a constant. [This is not very physically reasonable, but is easier
for calculations.]

(a) If the total charge is Q o, calculate C.

(b) Find an expression for the electric field at R = 2R o.
(c) Find an expression for the electric field at R = R o/2.

101. W hite light is perpendicularly incident on a thin piece of glass

(n = 1.50) as shown. Interference maxima are found for light of
81 = 550.0 nm and 82 = 497.6 nm, among others. There are no maxima
between these two wavelengths.

(a) Calculate the thickness of the glass.

(b) Find the next shorter wavelength (less than 497.6) that will show constructive interference.
102. The current in the inductance is positive when in the direction shown. The
current is given by

(a) Calculate T. L = 3.15 mH

(b) Determine the maximum voltage on the capacitor.
C = 475 pF
(c) Find the total energy in the oscillations.
Io = 1.32 A
103. W e model the current distribution in a cylindrical bolt of lightning using the current
density

for R < Ro where R o = 7.20 cm.

(a) If j(R o) = 0, find the value of ".

(b) If the total current is 30,000 Amperes, calculate j o (numerical value).
(c) Calculate the magnitude of the magnetic field at an arbitrary value of R inside the bolt (R < Ro)..

104. (a) Find the angular position in degrees of the second order maximum for green light of 8 = 510 nm
from a diffraction grating with 8500 lines/cm.
(b) The third minimum from the center in a single-slit diffraction pattern with red light of 8 = 640 nm
is found at 3.25 cm from the center on a screen 4.50 m from the slit. Calculate the width of the slit.

(c) Electrons are accelerated from rest through a potential difference of 5500 volts. Calculate their
DeBroglie wavelength.
(d) Calculate the energy (in joules) of a photon of yellow light whose wavelength is 680 nm.
(e) UV light of wavelength 8 = 350 nm is incident on a metal surface. The maximum energy of
photoelectrons is found to be 3.20 × 10 -19 J. Calculate the work function for this metal.
(f) Calculate the polarizing angle (Brewster angle) for light incident on diamond (n = 2.42) which is
immersed in water (n = 1.33).

105. The numerical values for the circuit elements in this circuit are given
below the diagram.

(a) Find the current in R 1

(b) Find the current in R 4.
(c) Find the current in R 7.

g = 325 V; R 1 = 2750 S; R 2 = 1200 S; R 3 = 1500 S;

R 4 = 1300 S; R 5 = 2500 S; R 6 = 4700 S; R 7 = 3200 S

106. Given a long thin wire with a uniform charge density 8, of -5.25 × 10 -5 C/m.

(a) Calculate the electric field at P a distance of 3.25 cm from the

center of the wire.
(b) Calculate the magnitude of the potential difference V(Q) - V(P) if
Q is a distance 14.75 cm from the center of the wire.
(c) W hat is the sign of the potential difference V(Q) - V(P)? Explain.
(d) Calculate the energy stored in a cylindrical region of space
centered on the wire and with inner radius 3.25, outer radius 14.75
and length 2.00 m.
107. For the circuit shown assume the inductor is ideal. The switch is open for a long time and
closed at t = 0.

(a) Calculate the current in the inductor after 1.50 time constants have elapsed.
(b) Calculate the current in R 2 after 0.75 time constants have elapsed.
(c) The switch is closed for 2.00 time constants and then opened. Obtain an
expression for the current in the inductor with all numerical quantities evaluated.
Make t = 0 in this expression the point at which the switch is opened.
(d) W hen the switch is originally closed at t = 0, obtain an expression for the current
in R 1 as a function of time, with all quantities, except the time constant,
numerically evaluated.

g = 175 V; R 1 = 4175 S; R 2 = 3500 S; L = 0.750 mH

108. Initially all capacitors are discharged.

(a) If the switch is closed at t = 0, calculate the current in R as a function of

time, evaluating all numerical constants.
(b) After being closed a long time, the switch is opened, and then a
dielectric with 6 = 3.25 is inserted in C 2. Calculate the charge and
potential on each capacitor after this event.
(c) After (b) the switch is closed again. How much additional charge flows
from the battery to the capacitors if you wait a long time?

g = 375 V; C 1 = 625 pF; C 2 = 325 pF; C 3 = 400 pF; R = 2.30 × 10 4 Ohms

109. A bolt of lightning strikes the ground at point A near a tall metal fence. Assume the bolt is straight and
vertical.

(a) If the peak current is 33,000 A, what is the peak

magnetic field at point B?
(b) If the current rises from zero to its peak in 0.35 ms,
what is the current induced in a square of the metal
fence that is 2.00 m on a side. The resistance around
the square is 0.15 S. Assume the current increase is
linear.
(c) If the bolt has a diameter of 5.20 cm, and a current
distribution given by j(R) = j o(1 - "R 2) where R is
the distance from the center of the bolt, calculate " for the peak current.
(d) W ith the assumptions in (c), calculate j o for the peak current.

110. Two optically flat, thick, glass plates with index of refraction n = 1.55, are
set up as shown. At one edge the plates touch, and at the other edge they
are separated by placing a hair between them. At the position of the hair
there is bright interference band in reflection for 8 = 550 nm (yellow) and
for 506 nm (green). There is no other place between the hair and the
touching edge where the green and yellow maxima exactly coincide.
There are no other wavelengths between 506 and 550 that give an
interference maxima at the position of the hair.

(a) How thick is the hair?

(b) Calculate the next wavelength, shorter than 506 nm, that will also show a maximum at the position
of the hair?
111. (a) A two slit interference experiment is done using yellow light of 8 = 595 nm. The maxima on a
screen that is 4.37 m away from the slits are 1.75 cm apart. Calculate the slit separation.
(b) In a single slit experiment using yellow light of 8 = 595 nm, the second minimum is at 2 = 0.160°.
Calculate the width of the slit.
(c) Electrons are accelerated from rest through a potential difference of 6500 V. Calculate their
de Broglie wavelength.
(d) Blue light of 8 = 450 nm is incident on a metal surface. The maximum kinetic energy found for
the electrons emitted is 1.75 electron Avolt. Calculate the work function for this metal.
(e) A diamond (n = 2.40) is immersed in water (n = 1.33). Calculate the critical angle for total internal
reflection.
(f) Light is incident on a 60-60-60 prism (n = 1.50) in air at 35°
from the normal. Calculate the angle 2 between the incident
and emergent rays.

112. Given the charge distribution shown.

(a) Find the magnitude of the electric field at P.

(b) Find the direction of the electric field at P, as an angle measured from the
positive x-axis.
(c) Calculate the electric potential at P, including sign, with usual choice of
V = 0.

Q 1 = +6.00 :C; Q 2 = !8.00 :C; a = 3.65 cm

113. A non-conductor, in the shape of a sphere of radius R o, is positively charged with an electric charge
distribution given by

(a) If the total charge is Q o, obtain an expression for Do in terms of Q o

and R o.
(b) Calculate the electric field in the interior of the sphere at a point
R = R o/3.
(c) Determine the electric potential at the point R o/3, using the usual assumption
about the location of V = 0.

114. In the circuit S 1 is closed and then opened. After S 1 is opened, S 2 is closed.

(a) Calculate the potential and charge on each capacitor in the final
configuration.
(b) How much energy is lost when S 2 is closed?

g = 135 V; C 1 = 17.0 pF; C 2 = 8.00 pF; C 3 = 12.0 pF; C 4 = 32.0 pF

115. For the wire shown, the current is given by I = I o sin Tt. All elements are in the plane of the paper. Positive
current is to the right. Here T = 325 rad/s. The loop below the wire is a conductor with the total resistance
shown.

(a) If the maximum value of the magnetic field at a distance a = 6.50 cm away from the wire is 27.0
gauss, what is the value of I o?
(b) Calculate the peak value of the current in the rectangular
conducting loop, using the result from (a).
(c) At t = 0, is the current in the loop going clockwise or counter
clockwise? Explain.
(d) Calculate the maximum value of the energy density in the
magnetic field at a point 2a away from the straight wire. Use the
result from (a). Numerical value.
116. For the circuit shown the switch S 1 is closed for a long time and then
opened. Switch S 2 is closed immediately after S 1 is opened. Switch S 2
is closed at t = 0.

(a) Calculate the frequency of oscillation (in Hz).

(b) Calculate the magnitude of the maximum current in the
inductor.
(c) Calculate the energy stored in the inductor at Tt = B/6, where
T is the angular frequency of oscillation.

g = 165 V; C = 18.5 :F; L = 23.0 mH

117. A beam of light is incident from medium Î to medium Ï in the drawing. The
index of refraction of medium Î is 1.20.

(a) Calculate the index of refraction for medium Ï.

(b) Find the critical angle for total internal reflection at the boundary.
(c) Determine the polarizing angle for light incident from medium Î.
(d) Calculate the speed of light in medium Î.

118. (a) The wire shown carries a current of I = 2.75 A.

Calculate the magnetic field at A, which is at the center
of the semi-circular part (r = 3.17 cm).
___________________________________

(b) Completely unpolarized light is incident on 3 polarizers. The

polarization axes of A and B make an angle of 30°. The
polarization axes of B and C have an angle of 48°. Calculate the
output intensity as a fraction of the initial intensity, I o.
__________________________________

(c) Find the critical angle for total internal reflection for a glass-water interface.
(n glass = 1.65, n water = 1.33) _____________________________________________________
(d) W hat is the de Broglie wavelength for electrons that have been accelerated through a potential
difference of 630 V? _________________________________________________________
(e) Light of wavelength 8 = 320 nm is incident on a metal surface. The maximum energy of the
photoelectron observed is 2.75 × 10 -19 J. W hat is the value of the work function for this material
(in J)? _____________________________________________________________________
(f) W hat is the energy (in eV) of a photon of green light ( 8 = 500 nm)? _____________________

119. (a) Calculate the time constant for the increase in current after the switch is
closed. g = 150 V; L = 225 mH; R = 1750 S
_________________________________________

(b) Calculate the inductance of an ideal solenoid 2.50 m long, 1.25 cm in diameter, with 17,700 total
turns of wire. ______________________________________________________________
(c) Calculate the magnitude of the electric field a distance of 1.10 × 10 -13 m away from the nucleus of
an hydrogen atom. _________________________________________________________
 (d) Calculate the position, with appropriate sign, of the image in this
system, measured from the center of the lens. ______________

(e) Calculate the polarizing angle for light reflected from diamond
(n = 2.40) that is immersed in water (n = 1.33). ____________
(f) Calculate the radius of the circular path of an electron if it was accelerated by a potential difference
of 15.0 V, and there is a magnetic field of 0.015 T perpendicular to its motion. _____________

120. Given a spherically symmetric distribution of electric charge, where the charge distribution is given as
D = Do R 4 for R # R o and 0 for R > R o.

(a) If the total charge is Q o, calculate Do.

(b) Calculate the electric field at R = R o/7.
(c) Calculate the magnitude of the electric potential difference between R = R o/7 and R = R o.

121. Yellow light ( 8 = 589 nm) is normally incident on a single slit whose width
is 0.112 mm. The light is projected on a screen d = 4.35 m away from the slit.

(a) Calculate the distance on the screen between the first dark minima on
the left and the second minima on the right.
(b) Blue light ( 8 = 504.9 nm) is also incident on the same slit with the
same geometry. Calculate the position on the screen where minima
of the two colors first overlap. Measure this from the center of the
diffraction pattern.

122. For the circuit shown all capacitors are initially uncharged.
The switch S 1 is closed at t = 0. and left closed for a long

time. S 2 is open.

(a) Calculate the time constant for charging this system.

(b) S 1 is opened and then S 2 is closed. Determine the
charge and potential on each capacitor after a long
time.
(c) A dielectric with 6 = 2.50 is inserted in C 2. There is
no change in the switches. Calculate the new
potentials and charges in all capacitors.

C 1 = 25.0 :F R = 737 S
C 2 = 37.0 :F g = 135 V
C 3 = 51.0 :F

123. For the circuit shown the switch is open for a long time, and closed at
t = 0.

(a) W hat is the current in R 3 at t = 0?

(b) W hat is the current in R 3 at t = 4?
(c) Calculate the current in R 2 after 1.20 time constants have
elapsed from t = 0.
(d) Determine the numerical value of the time constant for
discharge when the switch is opened.
(e) Using full loop and junctions and no shortcuts learned in other
classes, calculate the time constant (numerical value) for charging beginning at t = 0.

R 1 = 275 S g = 175 V
R 2 = 150 S L = 37.5 mH
R 3 = 350 S
124. The wire shown has a current given by I = I o e -kt (it is charging
a capacitor?). k is a positive constant.

(a) At t = 1/k calculate the flux through the rectangular

wire shown. All parts are in the plane of the paper.
(b) Calculate the magnitude of the voltage across the
resistor as a function of time.
(c) Is the current in the rectangular wire clockwise or
counter clockwise? Explain.
Data: Use these constants (where it states, for example, 1 ft, the 1 is exact for significant figure purposes).

1 ft = 12 in (exact)
1 m = 3.28 ft
1 mile = 5280 ft (exact)
1 hour = 3600 sec = 60 min (exact)
1 day = 24 hr (exact)
gearth = 9.80 m/s2 = 32.2 ft/s2
gmoon = 1.67 m/s2 = 5.48 ft/s2
1 year = 365.25 days
1 kg = 0.0685 slug
1 N = 0.225 pound
1 horsepower = 550 ft@pounds/s (exact)
Mearth = 5.98 × 1024 kg
Rearth = 6.38 × 103 km
Msun = 1.99 × 1030 kg
Rsun = 6.96 × 108 m
Mmoon = 7.35 × 1022 kg
Rmoon = 1.74 × 103 km
G = 6.67 × 10-11 N@m2 /kg2
k = 9.00 × 109 N@m2 /C2
go = 8.85 × 10-12 F/m
eelectron charge = -1.60 × 10-19 C
melectron = 9.11 × 10-31 kg
:o = 4B × 10-7 T@m/A (exact)
N(Avogadro's No.) = 6.02 × 1023 atoms/gm @mole
= 6.02 × 1026 atoms/kg@mole
1 Tesla = 10,000 gauss (exact)
D(H2 O) = 1000 kg/m3
cos(a ± b) = cos a cos b K a sin b
sin(a ± b) = sin a cos b ± sin b
mproton = 1.67 × 10!27 kg
B = 3.14