Constant Electromagnetic Field

1. Constant Electromagnetic Field Section 19

2. Constant fields • E and H are independent of time t. • f and Acan be chosen time independent, too.

3. We can add an arbitrary constant to f without changing E or H. • Only a constant (no t or r dependence) can be added to f. • An extra condition is usually imposed, e.g. f = 0 at infinity. • Then fis determined uniquely.

4. We can add an arbitrary constant to Awithout changing E or H, but we can also add functions. • A function of coordinates grad(f) can still be added to A without changing E or H. • A is not unique even for constant fields.

5. Energy of charge in constant electromagnetic field. • If fields are constant, the Lagrangian is independent of time. • The energy is conserved and equals the Hamiltonian.

6. The constant field adds energy ef to the particle. • ef is the “potential” energy of a charge in the field. • The energy does not depend on A, so H does no work on the charge. • Only E changes the energy of a particle.

7. Uniform constant fields • Electric field has no r dependence • f = -E.r

8. Uniform constant fields • A is not unique • Two examples that both give uniform H: A1 = (1/2) H x r A2 = [0,0,H] • These two choices differ by grad(f), where f = -xyH/2

12. Which function f(x) vs. x does not give a constant uniform electric field? 1 2 3

13. Which function f(x) vs. x does not give a constant uniform electric field? 3