1. Maxwell’s Equations (so far…)
Gauss’ Law for electric fields
Gauss’ Law for magnetic fields
Faraday’s Law
*Not complete

2. Something is missing in Ampere’s Law

3. A capacitor, connected to a circuit
Side view

4. A capacitor, connected to a circuit
Side view

5. A capacitor, connected to a circuit
Side view

6. These answers can’t both be right!
Something missing here…
These answers can’t both be right!

7. Faraday found that time-varying magnetic fields are accompanied by electric fields.
Maybe time-varying electric fields are also accompanied by magnetic fields?...
James Clerk Maxwell
(1831 – 1879)

8. The electric field between the plates is changing in time, as the capacitor is charged.
Field inside a capacitor:
Electric flux:

9. The electric field between the plates is changing in time, as the capacitor is charged.
Field inside a capacitor:
Electric flux:

10. The electric field between the plates is changing in time, as the capacitor is charged.
Field inside a capacitor:
Rate of change of the flux:
(since i = dq/dt)

11. The electric field between the plates is changing in time, as the capacitor is charged.
Field inside a capacitor:
Result:

12. This is a relationship between
the changing electric flux inside the capacitor, and
the current outside the capacitor.

13. Maxwell used it to fix Ampere’s Law:

14. Maxwell used it to fix Ampere’s Law:

15. Ampere-Maxwell Law
This is the last of Maxwell’s equations.

16. Now the answers are the same.

17. Maxwell’s Equations (so far…)
Gauss’ Law for electric fields
Gauss’ Law for magnetic fields
Faraday’s Law
*Not complete

18. Maxwell’s Equations Gauss’ Law for electric fields
Gauss’ Law for magnetic fields
Faraday’s Law
Ampere-Maxwell Law

19. Maxwell’s Equations in empty space
No charge: q = 0 No currents: i = 0

20. Maxwell’s Equations in empty space
Even with no charges or currents nearby, time-varying electric and magnetic fields can exist… as travelling waves.

21. Maxwell’s equations predict travelling waves.

22. Maxwell’s equations predict travelling waves.

23. Maxwell’s equations predict travelling waves.

24. Maxwell’s equations predict
light.