1. Time-dependent fields

2. Static fields decoupled
.D = r
 x E = 0
.B = 0
 x H = J
D = eE
B = mH
Electric fields have zero curl
Caused by Static Charges
Magnetic fields have zero divergence
Caused by Static Currents
Asymmetry in E and B field properties

3. 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)

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

5. Gauss’ Law for electrostatics (Flux prop. to enclosed charge)
The 4 Maxwell equations
.D = r
Gauss’ Law for electrostatics
(Flux prop. to enclosed charge)

6. Gauss’ Law for magnetostatics (There is no magnetic charge)
The 4 Maxwell equations
.B = 0
Gauss’ Law for magnetostatics
(There is no magnetic charge)

7. Faraday’s law of induction (Changing magnetic flux
The 4 Maxwell equations
 x E = - ∂B/∂t
Faraday’s law of induction
(Changing magnetic flux
creates voltage)

8. (Changing electric flux creates magnetic field)
The 4 Maxwell equations
 x H = J + ∂D/∂t
Ampere’s law
(Changing electric flux
creates magnetic field)

9. The 4 Maxwell’s equations
.D = r
 x E = - ∂B/∂t
.B = 0
 x H = J + ∂D/∂t
Not just E,B but D,H and also inputs r, J = sE
B = mH
D = eE
Constitutive equations
(Maxwell’s eqns don’t give e, m, s, r
Need quantum mechanics/solid state/statistical physics for this!)
We treat them as external inputs

10. The 4 Maxwell’s equations
.D = r
 x E = - ∂B/∂t
.B = 0
 x H = J + ∂D/∂t
Most important consequence: Electromagnetic Waves
(Chapter 7)
Here we’ll learn the two new equations
(Faraday’s Law and Ampere’s Law)

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

12. Faraday’s Law
 x E .dA = - ∂B.dA/∂t
Integrate both sides

13. Faraday’s Law
E .dl = - ∂FB/∂t
Stokes Theorem

14. Changing magnetic flux
Faraday’s Law
E .dl = - ∂FB/∂t
Changing magnetic flux
creates voltage

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

16. Changing magnetic flux
Faraday’s Law
 x E = - ∂B/∂t
Changing magnetic flux
creates voltage
(and thus current)

17. How to change magnetic flux?

18. Lenz’s Law Induced current opposes any change in flux
(Nature Prefers Inertia)
opposing induced flux
Increasing flux

19. Lenz’s Law Induced current opposes any change in flux
(Nature Prefers Inertia)
opposing induced flux
Decreasing flux

20. Force argument for Lenz’s Law
Lorentz Force
-e(v x B) downwards on el
Wire motion
Current flows
upward in slider
Current flows
upward in slider
Current flows
upward in slider
Decreasing B Flux towards you
 induced B also towards you
Increasing B Flux towards you
 induced B away from you

21. No matter which way you see it,
the current in the slider flows the
same way !!

22.
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/lenzlaw/
/transformer
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