1. Time-dependent fields

2. Maxwell’s equations for Electromagnetism
Dynamic fields coupled
.D = r
 x E = - ∂B/∂t
.B = 0
 x H = J + ∂D/∂t
D = eE
B = mH
Maxwell’s equations for Electromagnetism
Field equations more symmetric
(fields resemble each other)

3. Deriving Circuit Theory !
.D = r
 x E = - ∂B/∂t
.B = 0
 x H = J + ∂D/∂t
Kirchhoff’s Law
V=LdI/dt

4. Changing magnetic flux
Faraday’s Law
 x E = - ∂B/∂t
Since  x E ≠ 0 , can’t have E = -U
Changing magnetic flux
creates voltage

5. Faraday’s Law
Vemf = - ∂FB/∂t
Definition: FB/I = L
(Solenoid)

6. How to change magnetic flux?

7. Transformer: An electrical ‘gear’
Automobile Ignition Coil
Vp=NpdFp/dt
Vs=NsdFs/dt
Fp=Fs
Vs/Vp = Ns/Np
P1=P2
Zs/Zp = N2s/N2p

8. Wikipedia/howstuffworks
This is how a car runs
Large Voltage (40 KV) ionizes air in spark
gap and creates high temperature (60000K)
that causes dielectric breakdown and
creates combustion of fuels
The explosion drives a 4-stroke cycle
engine that drives the car
Wikipedia/howstuffworks

9. Eddy Currents
UMD Physics LecDems

10. Wikipedia/howstuffworks
Metal Detectors
Incident pulse in coils cause a magnetic field to reverse and
collapse. A reflected pulse is created by the collapse. Any
induced magnetic field in metals delays collapse of reflected
pulse by a few ms, which is detected.
Wikipedia/howstuffworks

11. Electromagnetic Potential
.B = 0  B =  x A
 x E = - ∂( x A)/∂t

12. New relation between E and V
Electromagnetic Potential
.B = 0  B =  x A
x (E + ∂A/∂t) = 0
E + ∂A/∂t = - V
E = -V - ∂A/∂t
New relation between E and V

13. Equation of Current Continuity
.D = r
 x H = J + ∂D/∂t

14. Charging a Capacitor

15. Equation of Current Continuity
.D = r
 x H = J + ∂D/∂t

16. Equation of Current Continuity
.∂D/∂t
= ∂r/∂t
 x H = J + ∂D/∂t

17. Equation of Current Continuity
.( x H – J)
= ∂r/∂t
 x H = J + ∂D/∂t

18. Equation of Current Continuity
.J = -∂r/∂t

19. Equation of Current Continuity
.J dv = -∂rdv/∂t

20. Equation of Current Continuity
J.dA = -∂Q/∂t

21. Equation of Current Continuity
I = -∂Q/∂t

22. Modified Ampere’s Law
 x H = J + JD

23. Equation of Continuity
0 = .(J + JD)
Displacement Current
Kirchhoff’s Current Law

24. Charging a Capacitor + - + - I + - + - + -
How do we visualize displacement currents?
As charges crowd onto plates,
E increases between plates
These ‘link’ increasing + and –
charges on plates and act like a
current that bridges them + - + - I + - + - + -
This is the displacement current
JD = ∂D/∂t

25. The 4 Maxwell’s equations
.(eE) = r
 x E = - ∂B/∂t
.B = 0
 x (B/m)= J + ∂(eE)/∂t sE
Consequences:
Electromagnetic Waves (Ch 7)
that propagate radiatively for oscillating charges (Ch 9)
and bend in specific ways at interfaces (Ch 8)