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.

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.

Maxwell’s Equations Integral Form Differential Form

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

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}

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

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

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

Two of Maxwell’s Equations Nambla: a vector derivative

Two of Maxwell’s Equations Nambla: a vector derivative

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)

Using Maxwell’s Equations

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