ELECTROSTATICS

1 𝑞1 𝑞2
Force = 𝐹 = 4𝜋𝜀
0 𝑟2
𝐹
Electric field intensity 𝐸 = 𝑞

Electric dipole moment 𝑝 = 𝑞 × 2𝑙

1 2𝑝
Intensity at a point on axial line 𝐸 = 4𝜋𝜀
0 𝑟3
1 𝑝
Intensity at a point on Equatorial line = 𝐸 =
4𝜋𝜀0 𝑟 3

Torque acting on a dipole 𝜏 = 𝑝𝐸 𝑠𝑖𝑛𝜃

Electric flux = 𝜑 = 𝐸𝐴 cos 𝜃
𝑞
Gauss’ theorem 𝜑 = 𝜖
0

𝜆
Electric field due to an infinite line charge𝐸 = 2𝜋𝜀0 𝑟
𝜎
Electric field due to an infinite plane sheet of charge 𝐸 = 2𝜀0

Electric field due to charged thin shell

1 𝑞
a) At an external point𝐸 =
4𝜋𝜀0 𝑟 2
𝜎
b) At the surface 𝐸 = 𝜀0
c) At an internal point E =0
ELECTRIC POTENTIAL
𝑤
𝑉=
𝑞0
1 𝑞
Potential due to a point charge 𝑉 = 4𝜋𝜀 𝑟
0
 1 𝑝
Potential due to a dipole at a point on axial line 𝑉 4𝜋𝜀
0 𝑟2

Potential due to a dipole at a point on equatorial line = 0

𝑑𝑉
Relation between E and V 𝐸 = − 𝑑𝑟
1 𝑞1 𝑞2
Electric potential energy 𝑈 = 4𝜋𝜀
0 𝑟

Work done in rotating an electric dipole 𝑊 = 𝑝𝐸 (1 − 𝑐𝑜𝑠𝜃 )

Potential energy of diploe – 𝑈 = −𝑝𝐸 𝑐𝑜𝑠𝜃

CAPACITANCE
𝑄
Capacitance 𝐶 =
𝑉

Capacitance of a spherical conductor 𝐶 = 4𝜋𝜀0 𝑟

𝜀0𝐴
Capacitance of a parallel plate capacitor 𝐶 = 𝑑
𝜀0𝐴
Capacitance with a dielectric between the plates 𝐶 = 𝑡
𝑑−𝑡 + 𝐾

1 1 1 1
Capacitors in series𝐶 = + 𝐶 + 𝐶 ………
𝐶1 2 3

Capacitors in parallel 𝐶 = 𝐶1 + 𝐶2 + 𝐶3 … … ..
𝐶𝑉 2 𝑄2
Energy stored in a capacitor𝑈 = =
2 2𝐶
𝜀
Dielectric constant K =𝜀𝑟 = 𝜀
0

CURRENT ELECTRICITY
Current in terms of drift velocity 𝐼 = 𝑛𝑒𝐴𝑣𝑑
𝐼
Current density𝑗 = 𝐴 = σE
 𝐸 𝑅𝐴
Specific resistance 𝜌 = 𝑗 = 𝑙

𝑣2𝑡
Electric Energy 𝑊 = 𝐼2 𝑅𝑡 = = 𝑉𝐼𝑡
𝑟

2 𝑣2
Electric power 𝑃 = 𝐼 𝑅 = = 𝑉𝐼
𝑟
1 1 1 1
Resistors in parallel𝑅 = + 𝑅 + 𝑅 ………
𝑅1 2 3

Resistors in series 𝑅 = 𝑅1 + 𝑅2 + 𝑅3 … … ..
𝑅𝑡 −𝑅0
Temperature coefficient of resistance𝛼 = 𝑅0 𝑡

Terminal potential difference𝑉 = 𝐸 − 𝐼𝑟

𝑛𝐸
Cells in series 𝑖 = 𝑛𝑟+𝑅
𝑛𝐸
a) When nr<<R ,𝑖 = 𝑅
𝐸
b) When nr>>R, 𝑖 = 𝑅
𝑛𝐸
Cells in parallel 𝑖 = 𝑟+𝑛𝑅
𝐸
a) When nr<<R ,𝑖 = 𝑅
𝐸
b) When nr>> R, 𝑖 = 𝑛 𝑅
𝑃 𝑅
Wheatstone bridge balancing condition 𝑄 = 𝑆

100−𝑖
Metre Bridge Unknown resistance 𝑆 = 𝑅 ( )
𝑙

𝑉𝐴𝐵
Potential gradient 𝐾 = , smaller the value of K grater will be the
𝐿
sensitivity.
𝑙
Internal resistance𝑟 = 𝑅 ( 𝑙1 − 1)
2
MAGNETISM AND MOVING CHARGE
0 𝜇 𝐼
Magnetic field due to infinitely long conductor 𝐵 = 2𝜋𝑟
𝜇0𝑁 𝐼
Magnetic field due to circular coil (at the center) 𝐵 = 2𝑟
𝜇0𝑛 𝐼
Magnetic field due to a solenoid 𝐵 = ( n= N/L)
𝜇0𝑁 𝐼
Magnetic field due to a Toroidal 𝐵 = 2𝜋𝑟

Force acting on a moving charge 𝐹 = 𝐵𝑞𝑣𝑠𝑖𝑛𝜃

Force acting on a current carrying conductor 𝐹 = 𝐵𝐼𝑙𝑠𝑖𝑛𝜃
𝜇0 𝐼1 𝐼2
Force between two parallel current carrying conductor𝐹 == 𝑙
2𝜋𝑟

Torque acting on a current carrying loop 𝜏 = 𝑁𝑖𝐴𝐵𝑠𝑖𝑛𝜃

Magnetic dipole moment 𝑚 = 𝑁𝑖𝐴
𝜑 𝑁𝐴𝐵
Current sensitivity of moving coil galvanometer 𝑖 = 𝐶
𝜑 𝑁𝐴𝐵
Voltage sensitivity of galvanometer 𝑉 = 𝐶
𝐼𝑔
Galvanometer into ammeter value of shunt resistance 𝑆 = (𝐼−𝐼 ) 𝐺
𝑔

𝑉
Galvanometer into voltmeter value of series resistance 𝑅 = −𝐺
𝐼𝑔

Magnetic field due to a magnetic dipole

𝜇 2𝑚
(a) End on position 𝐵 = 4𝜋0 𝑟3
𝜇0 𝑚
(b) Broad on position 𝐵 =
4𝜋 𝑟 3

Torque acting on a magnetic dipole 𝜏 = 𝑚𝐵 𝑠𝑖𝑛𝜃

Horizontal component of earth’s magnetic field 𝐵𝐻 = 𝐵𝐸 𝑐𝑜𝑠𝜃
Vertical component of earth’s magnetic field 𝐵𝑉 = 𝐵𝐸 𝑆𝑖𝑛𝜃
𝐵𝑉
= 𝑡𝑎𝑛𝜃
𝐵𝐻

𝐵𝐸 = √𝐵𝐻 2 + 𝐵𝑉 2

MAGENTIC SUBSTANCES
𝑚
Intensity of magnetization 𝑀 =
𝑉
𝐵
Magnetic field strength 𝐻 = 𝜇 − 𝑀
0

𝐵
Magnetic permeability𝜇 =
𝐻
𝜇
Relative magnetic permeability 𝜇𝑟 = 𝜇
0

𝑀
Magnetic Susceptibility 𝜒 = 𝐻

Relation between 𝜇𝑟 𝑎𝑛𝑑 𝜒𝜇𝑟 = 1 + 𝜒

ELECTROMAGENTIC INDUCTION
𝑑∅
Induced emf𝑒 = − 𝑑𝑡

Motional emf𝑒 = 𝐵𝑣𝑙

𝑑𝑖
Self- induced emf𝑒 = −𝐿
𝑑𝑡
𝜇0 𝑁 2 𝐴
Self-inductance of a long solenoid 𝐿 = 𝑙
𝑒2
Coefficient of mutual inductance𝑀 = − 𝑑𝑖
𝑑𝑡

𝜇0 𝑁1 𝑁2 𝐴
mutual inductance (two co axial solenoids) 𝑀 = 𝑙
Induced voltage in an ac generator 𝑒 = −𝑁𝐴𝐵𝜔𝑆𝑖𝑛𝜔𝑡
Mean value of ac current𝐼𝑚 = 0.637 𝐼0
RMS value𝐼𝑟𝑚𝑠 = 0.707 𝐼0
AC circuit with R only 𝐼 = 𝐼0 𝑆𝑖𝑛𝜔𝑡
𝜋
AC circuit with L only 𝐼 = 𝐼0 𝑆𝑖𝑛(𝜔𝑡 − 2 )
𝜋
AC circuit with C only 𝐼 = 𝐼0 𝑆𝑖𝑛(𝜔𝑡 + 2 )

Inductive reactance 𝑋𝐿 = 2𝜋𝑓𝐿

1
Capacitive reactance 𝑋𝐶 = 2𝜋𝑓𝐶

Impedance of LCR circuit𝑍 = √𝑅 2 + (𝑋𝐿 − 𝑋𝐶 )2

𝑋𝐿 −𝑋𝐶
Phasetan ∅ = 𝑅

Power in LCR circuit 𝑝 = 𝑉𝑟𝑚𝑠 × 𝐼𝑟𝑚𝑠 𝑐𝑜𝑠∅

1
Resonance frequency of LCR circuit 𝑓 = 2𝜋√𝐿𝐶
𝑖𝑝 𝑉 𝑁
Transformer equation 𝑖 = 𝑉𝑠 = 𝑁 𝑠
𝑠 𝑝 𝑝

ELECTROMAGNETIC WAVES
𝑑∅𝐸
Displacement current 𝐼𝑑 = 𝜀0 𝑑𝑡
𝑅
Focal length 𝑓 = 2
1 1 1
Mirror formula +𝑢=𝑓
𝑣
𝐼 𝑣
Linear magnification 𝑚 = =
𝑂 𝑢
 𝑓−𝑣
𝑚=
𝑓
𝑓
𝑚=
𝑓−𝑢
sin 𝑖
Snell’s law 𝜇21 =
sin 𝑟

𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑖𝑛 𝑚𝑒𝑑𝑖𝑢𝑚 1
𝜇21 =
𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑖𝑛 𝑚𝑒𝑑𝑖𝑢𝑚 2
1
𝜇21 =
𝜇12
𝑅𝑒𝑎𝑙 𝑑𝑒𝑝𝑡ℎ
𝜇21 =
𝑎𝑝𝑝𝑎𝑟𝑒𝑛𝑡 𝑑𝑒𝑝𝑡ℎ
1
𝜇21 =
sin 𝐶
𝑛2 𝑛1 𝑛2 −𝑛1
Refraction at spherical surface − =
𝑣 𝑢 𝑅
1 1 1
Refraction through lens 𝑓 = (𝑛 − 1)(𝑅 − 𝑅 )
1 2

1
Power 𝑝 = 𝑓(𝑖𝑛 𝑚𝑒𝑡𝑒𝑟𝑠)
1 1 1
Focal lengths of lenses in contact 𝑓 = 𝑓 + 𝑓
1 2

Refraction through prism 𝑟1 + 𝑟2 = 𝐴

Angle of deviation 𝑑 = 𝑖1 + 𝑖2 − 𝐴
(sin 𝐴+𝑑𝑚 )/2
Refractive index 𝑛 = 𝑆𝑖𝑛 𝐴/2
𝑛𝑣 −𝑛𝑟
Dispersive power 𝜔 =
𝑛𝑦 −1

OPTICAL INSTRUMNETS
Simple microscope
𝐷
Magnification 𝑚 = + 1 (image at Least distance of vision)
𝑓
𝐷
𝑚 = 𝑓 (at infinity)

Compound microscope
𝐷 𝑣 𝐷 𝑓𝑜
𝑚 = (𝑓 + 1) 𝑢𝑜 or 𝑚 = (𝑓 + 1) 𝑢 ( Least distance of vision)
𝑒 𝑜 𝑒 𝑜 −𝑓𝑜

Length of the tube= 𝑣𝑜 + 𝑢𝑒

𝐷 𝑣
𝑚 = (𝑓 ) 𝑢𝑜 (at infinity)
𝑒 𝑜

Length of the tube= 𝑣𝑜 + 𝑓𝑒

𝑓
Telescope 𝑚 = 𝑓0(at infinity)
𝑒

𝑓𝑒 𝑓𝑜
𝑚=( + 1)
𝐷 𝑓𝑒
INTERFERENCE
Constructive interference Path difference= nλ (n=0,1,2…….)
Phase difference = m 2π (m=0,1,2,…….)
destructiveinterference Path difference= (2n-1)λ/2 (n=,1,2,3…….)
Phase difference = (2m-1) π (m=1,2,3…….)
𝐼𝑚𝑎𝑥 [𝑎1 + 𝑎2 ]2
=
𝐼𝑚𝑖𝑛 [𝑎1 − 𝑎2 ]2
𝜆𝐷
Fringe width𝛽 = 𝑑
𝜆
Angular distance between two fringes𝜃 =
𝑑
DIFFRACTION
Condition for minima 𝑑 𝑠𝑖𝑛𝜃 =nλ (n=1,2…….)
n= 0 corresponds to central maximum
Conditions for secondary maximum𝑑 𝑠𝑖𝑛𝜃 =(2n+1)λ/2 (n=,1,2,3…….)
𝜆
Angular width of central maxima 2𝜃 = 2𝑠𝑖𝑛−1 ( )
𝑑

POLARIZATION
𝜇 = 𝑡𝑎𝑛𝑖𝑝 Law of Malus 𝐼 ∞ 𝑐𝑜𝑠 2 𝜃
MODERN PHYSICS
Work function𝑊 = ℎ𝜗0
Kinetic energy 𝐾 = ℎ(𝜗 − 𝜗0
ℎ
Momentum 𝑝 = λ
h
De-Broglie wavelength of electronλ =
√2𝑚𝑒𝑉
1 2𝑍𝑒 2
Distance of closest approach for alpha particle𝑟0 = 4𝜋𝜀 𝐾
0

Bohr Model
mvr = nh/2π
𝐸2 − 𝐸1
𝜗=
ℎ
𝑛2 ℎ 2 𝜀0
Radius 𝑟 = (r ∞𝑛2
𝜋𝑚𝑍𝑒 2
 𝑍𝑒 2 1
Velocity𝑣 = 2ℎ𝜀 (𝑣 ∞ 𝑛)
0𝑛

𝑚𝑍 2 𝑒 4 1 −13.6
Energy𝐸 = 8𝜀 2 ℎ2 (𝑛 2 ) 𝐸 = eV
0 𝑛2

1 1 1
= 𝑅( 2 − 2
𝜆 𝑛1 𝑛2
𝑚𝑒 4
𝑅=
8𝜀0 2 ℎ3 𝑐
12375
𝜆= ∆𝐸
Angstrom

NUCLEUS
1
Nuclear radius 𝑅 = 𝑅0 𝐴 3

3𝑚
Density of nucleus𝜌 = 4𝜋𝑅 3
0

1 amu = 1.6605 X 10-27kg

N= 𝑁0 𝑒 −𝜆𝑡
0.693
𝑡ℎ𝑎𝑙𝑓 =
𝜆
𝑛
1
𝑁 = 𝑁0 ( )
2
1
Average life 𝜏 = 𝜆

Binding energy =(𝑍𝑚𝑝 + (𝐴 − 𝑍 )𝑚𝑝 − 𝑚 𝑜𝑓 𝐴𝑧𝑋 )C2

SEMICONDUCTORS
𝑣𝑖𝑛𝑝𝑢𝑡 −𝑣𝑜𝑢𝑡 𝑉𝑖𝑛𝑝𝑢𝑡 −𝑉𝑍
Zener Voltage regulator 𝑅 = =
𝐼 𝐼𝐿 +𝐼𝑍

Relation between transistor currents𝐼𝐸 = 𝐼𝐵 + 𝐼𝑐

 𝑖
Current gain𝛽 (𝑑𝑐) = 𝑖𝑐
𝑏

∆𝐼
Current gain𝛽 (𝑎𝑐) = ∆ 𝐼 𝐶 𝑓𝑜𝑟 constant VCE
𝐵

∆ 𝐼
Transconductance𝑔𝑚 = ∆ 𝑉 𝐶 𝑓𝑜𝑟 constant VCE
𝐵𝐸

𝛽 (𝑎𝑐)
Transconductance𝑔𝑚 = 𝑅𝑖𝑛
𝑅𝑜𝑢𝑡
Voltage gain 𝐴𝑣 = 𝛽 𝑅𝑖𝑛
𝑅𝑜𝑢𝑡
Power gain = current gain X voltage gain= 𝛽 2 𝑅𝑖𝑛