Maxwell's equations for electromagnetism in free space can be written as follows:
(i) ∇ · B = 0
(ii) ∇ · E = 0
(iii) ∇ × E + ∂B/∂t = 0
(iv) ∇ × B - (1/c²)∂E/∂t = 0
(a) The vector potential A is defined by B = ∇ × A and the scalar field ϕ by E = -∇ϕ - ∂A/∂t. Show that these relations are consistent with Maxwell's equations (i.e., that they satisfy Eqs. (i) and (iii)).
(b) By choosing the Lorentz gauge ∇ · A + (1/c²)∂ϕ/∂t = 0, show that ϕ satisfies the wave equation ∇²ϕ - (1/c²)∂²ϕ/∂t² = 0.
Hint: Substitute for E into (ii) and differentiate (v) with respect to time to eliminate A.
(c) Show that the vector potential A also satisfies the wave equation.
Hint: Substitute A and ϕ into the expressions for B and E in (iv). Then use the gradient of (v) to simplify the resulting expression.