E&M I
Griffiths Chapter 7
Example 7.9 Long straight wire with time dependent
current. What is the induced electric field?
changing
Recipe
1. Find B(t) using Ampere’s law
2. Find magnetic flux current –dF/dt
3. Find E(t) using Faraday’s law
Amperian loop
Step 1:
Ampere’s law:
Step 2: Flux current
Direction is consistent with
Lenz’s law
Step 3: Faraday’s law:
E(s,t)
But this is because wire was infinitely long
This was an example of a method called the Quasistatic approximation.
We used an equation from magneto-statics (Ampere’s law) to find B.
Then we used an equation from electrodynamics (Faraday’s law) to find E.
This approximation is valid if we can ignore “retardation”. That means that all
parts of space see the changes in current instantaneously.
That means we have to be sufficiently close to the wire, and the changes have
to be sufficiently slow:
Speed of light Characteristic time for changes in current
Because B1 is proportional to I1
Coefficient of proportionality, “Mutual Inductance”
2
Purely geometric, and symmetric under exchange 1 2.
Symmetry of mutual inductance has a practical benefit
Complicated loop 1 Simple loop 2
The symmetry of M12 means that the flux through loop 2 when there is current I in loop 1
is the same as the flux through loop 1 when the same current I is in loop 2.
Subscripts can be dropped M12 = M.
A change in I1, creates a change in B1 by Ampere’s law.
The change in B1 induces an EMF in loop 2 by Faraday’s law.
A change in I1 induces an EMF in any nearby loop, including in loop 1 (itself).
-dF1/dt “Self inductance”
Lenz’s law: self EMF opposes change in I1
Close the switch.
Current starts tries to start flowing.
“Back EMF” appears to oppose this current.
Practical examples of mutual and self inductance
Voltage transformer: Step up, step down.
Wire-wound resistor: No good for high frequencies.
Pulse transformer: Transmit voltage pulse without transmitting background voltage.
Bias-T: Separate DC power supply from high-frequency signals.
Etc, etc.
Magnetic energy.
Work is required to push current into an inductor against the back EMF.
d
= a voltage
= potential
= potential energy per unit
charge
Gravitational analogy
This work gets stored
dW = -F dx as potential energy
d
Total work done becomes an integral
over current, from 0 to I.
The work done depends only on geometry (L)
and on the final current I.
As for electric energy, there are different ways to express magnetic energy.
This integral is over the
volume of the wire, but
it can be extended to all
space considering that J
= 0 outside the wire.
Integral follows the
wire with current I By Ampere’s law
Product rule #6
div. Thm.
Magnetic Fields at infinite
energy boundary are zero
density
Example 7.10. Coaxial solenoids
1: Radius a, n1 turns per unit length
2: Radius b, n2 turns per unit length
If current = I in coil 1, what is magnetic
flux F2 through coil 2?
The field from coil 1 is non-uniform inside
coil 2, which makes this problem hard.
But, Symmetry of M!
Let current I flow in coil 2 instead. Its field
is uniform over coil 1.
So if I flows in coil 1,
Same M
I
Example 7.11. Toroidal coil.
N total turns of wire.
What is self inductance L?
Up arrows are currents in individual wires on
inner surface of toroid.
These are enclosed by Amperian loop, radius s.
Total enclosed current = NI
Total magnetic flux through all N
turns of toroid
= LI
Example 7.12. L-R circuit.
Close the switch. What is I(t)?
Kirchoff’s law
1st order inhomogenous diff. eq.
Initial condition
One undetermined coefficient
Example 7.13.
While there is current, the coax stores magnetic energy. How much?
Energy density, u =
Integrate over cylindrical shells
Radius s, thickness ds, length l
But So
Two recipes to find L.
1. Find flux for given I, then use F = LI
2. Find magnetic energy W for given I, then use W = (1/2) L I 2
The second method is a volume integral of a scalar, and it is usually the easier method.
Electrodynamics so far
Something missing
Problem:
Not
Mathematically zero necessarily
zero
Continuity equation
We can fix the problem
New Ampere’s law by adding the opposite
of this to Ampere’s law.
+
Maxwell’s correction: A
changing E can induce a B-field
Acts like a current: “The displacement current”
Old Ampere’s law and a charging capacitor
The perimeter is the
This current pierces same, but no usual
the surface bounded current penetrates the
by the loop surface
New Ampere’s law
Now any surface with the
same perimeter gives the
same answer
Basic equations to solve any Electrodynamics problem
1. Maxwell’s Equations for the fields
Gauss
Faraday
Ampere + Maxwell
2. Force law
Lorentz
3. Plus boundary conditions
Maxwell’s Equations
determine how
charges affect the
fields
Fields Charges
Lorentz force law tells
how fields affect the
charges
Fields
Charges
Maxwell’s equations would be symmetric if there were magnetic charges,
but there are not any.
Maxwell’s equations include all charges and all currents.
In matter, some are bound and some are free.
(I.e. some are intrinsic and some are extraneous.)
Bound charge
Bound current density
Polarization current density
Total charge
Total current density
Maxwell’s equations in terms of D and H and free charge
8 differential equations for 12 quantities E, B, D, H: Not enough equations!
“Constitutive Equations” are additional relations between E and D, H and B.
These depend on the medium.
For linear media
To obtain the needed boundary conditions, use integral form of Maxwell’s equations
Normal component of electric induction D is discontinuous by the surface free-charge density.
If there is none, then this component of D is continuous.
Normal component of magnetic induction B is always continuous across any boundary.
H1
Surface free H2 Amperian loop
current
density A/m
Goes to zero:
Displacement
current is
finite, but loop
is infinitesimal
or
Tangential components of H are discontinuous by free surface current
density components that flow perpendicular to them.
Loop
Goes to zero:
Magnetic flux
current is
finite, but loop
is infinitesimal
Tangential component of electric field is continuous
at a boundary
The total charge in some volume V is
If charge flows out of this volume through the
boundary surface
Div Theorem
Integrands have to be the same at
every point if equality holds for any
volume V
Continuity Equation
Some volume containing If a charge moves, then work
charges and currents and fields. is done on it by the field.
Rate at which work is done on
a single charge by the fields =
Rate at which work is done on all
the charges in V by the fields =
Power density delivered by the fields
Using Ampere-Maxwell Eq.
Product rule #6
Faraday’s law
Div Theorem
Poynting’s Theorem
Work done on Decrease in Rate of field
charges by fields field energy energy loss
per unit time per unit time through boundary
Energy per unit
Energy flux density
time per unit area
Energy flux Energy per unit time
Poynting’s Theorem
The work dW done on the charges increases their mechanical energy
Statement of Energy
Conservation
Cylindrical wire
with current I
Integrate over the cylindrical surface
The source of Joule
heat is field energy that
flows into the wire
from outside.
