Electricity and Magnetism Unit E5: Maxwell's Equations and Electromagnetic Waves

Prerequisite: E4, W1

Overview:

Maxwell's modification of Ampere's Law is an example of a new physical law found, not from experiment, but from carefully analyzing the inconsistency of the existing laws and finding a way to modify them so as to make them consistent. The modified laws had important consequences, which were verified by experiment. Most new physical laws are not found in this way, but when they are it is a triumph of human intellect.

Section I: Maxwell's Laws

E. M. Purcell, Electricity and Magnetism, Berkeley Physics Course, Vol. 2, 2nd Ed. , Chapt. 9 - Maxwell's Equations and Electromagnetic Waves, Sec. 9.1-9.3.

Understand:

1. Recall Gauss' laws for the electic flux and the magnetic flux through a closed surface S surrounding a volume V.
2. Recall Faraday's law relating the EMF around a closed loop to the negative of the rate of change of the magnetic flux through the loop.
3. Why Ampere's law relating the line integral of the magnetic field around a closed loop to the electric current throught the loop is not consistent with the above laws and therefore must be modified.
4. How one determines the modification and finds the extra term proportional to the rate of change of the electric flux through the loop which must be added to Ampere's law.

The two Gauss' laws, Farday's law, and the modified Ampere's law are known collectively as Maxwell's equations.

Maxwell's equations above are written in their "integral forms". You will probably find it useful to look at the equivalent "differential forms" in Eqn. 15 on page 330 of Purcell. In these equations the symbols "curl" and "div" are both linear in the three partial differential operators and operate on vector fields. The curl acts like a vector cross product and operating on a vector field C(r) produces another vector field: curl C = i(partial y C_z - partial z C_y ) + j (partial z C_x - partial x C_z ) + k(partial x C_y - partial y C_x ), where i, j and k are unit vectors in the x, y and z directions. In contrast the divergence (div) acts like a dot product and operating on a vector field produces a scalar field: div C(r) = partial x C_x + partial y C_y + partial z C_z. Of the four Maxwell's equations, then, the two curl equations are vector equations while the two divergence equations are scalar equations.

Section II: Electromagnetic Waves

E. M. Purcell, Electricity and Magnetism, Berkeley Physics Course, Vol. 2, 2nd Ed. , Chapt. 9 - Maxwell's Equations and Electromagnetic Waves, Sec. 9.4-9.6.

Understand:

1. For an electromagnetic wave travelling in the postitive y direction and polarized in the z direction the two fields (in CGS units) are E = k E_0 sin[(2pi/lambda)(y-ct)] and B = i E_0 sin[(2pi/lambda)(y-ct)] .
2. The wave carries energy in the direction in which it is travelling and at the speed of light. The time average energy per area per time is S = c/(4 pi) = E_0^2 c/(8 pi), since the time average of the square of a quantity varying sinusoidally is just half of its maximum value squared.

Problems:

Purcell 9.1, 9.2, 9.5, 9.10 .