1. Maxwell’s Equations
If we combine all the laws we know about electromagnetism, then
we obtain Maxwell’s equations.
These four equations plus a force law form the basis for
all of electromagnetism!
Thesbe laws predict that accelerating charges will radiate
electromagnetic waves!
The fact that classical models of the atom contradicted Maxwell’s
equations motivated quantum mechanics.

2. Maxwell’s Equations Integral Form
Gauss’s laws, Ampere’s law and Faraday’s law all combined!
They are nearly symmetric with respect to magnetism and electricity.
The lack of magnetic monopoles is the main reason
why they are not completely symmetric.

3. Maxwell’s Equations
Integral Form
Differential Form

4. Maxwell and Lorentz Force Law
Differential Form
FYI, These are connected to the integral equations via the generalized stokes equation

5. Derivatives and Partial Derivatives
When you have multiple variables, and you need to take the derivative, you use a partial derivative
Partial derivatives are like ordinary derivatives, but all other variables are treated as constants
{We have done this before; remember the gradient}

6. Vector Derivatives: Dot products in Cartesian Coordinates
Nambla: a vector derivative
is the divergence of B.

7. Vector Derivatives: Cross products in Cartesian Coordinates
Nambla: a vector derivative
is the curl of B.

8. Vector Derivatives: In other coordinates
Nambla needs to be converted
if we change coordinates
Spherical:

9. Two of Maxwell’s Equations
Nambla: a vector derivative

10. Two of Maxwell’s Equations
Nambla: a vector derivative

11. Waves from Electromagnetism
Consider electric fields (pointing in the y-direction) that depend only on x and t
Consider magnetic fields (pointing in the z-direction) that depend only on x and t
Consider vacuum , aka free space, so J=0
Plane waves
(We could be more general)

12. Using Maxwell’s Equations

13. Electromagnetic Waves
These equations look like sin functions will solve them.