A.1 Find the magnetic field B produced by a long straight wire carrying a steady current I, as a function of I and the distance r from the wire.
A.2 A physical electric dipole lying along the z-axis consists of charge +q at z = d/2 and charge -q at z = -d/2. Calculate the electrostatic potential φ(x) for large distances |x| ≫ d, as a function of q, d, |x|, and the angle θ between x and the z-axis. Show that the leading term for |x| ≫ d has the dipole potential form:
φ_dip = (1/4πε_0) * (p · x̂ / x²)
where p = qdẑ is the electric dipole moment. Using E = -∇φ, show that the electrostatic field due to the potential in (ii) is given by:
E_dip(x) = (1/4πε_0) * (3(p · x̂)x̂ - p) / |x|³
A.3 Consider the Maxwell equations in vacuum:
∇ · E = ρ/ε_0
∇ · B = 0
∇ ∧ E = -∂B/∂t
∇ ∧ B = μ_0J + μ_0ε_0∂E/∂t
Show that the two homogeneous Maxwell equations are automatically solved by the introduction of the electric and magnetic potentials φ and A via E = -∇φ - ∂A/∂t, B = ∇ ∧ A. Write down the two inhomogeneous Maxwell equations by eliminating E, B in favor of φ, A. Show that in the Lorentz gauge (c⁻²∂φ/∂t + ∇ · A = 0), the inhomogeneous equations take the form:
◻φ = -ρ/ε_0
◸A = -μ_0J
(◻ ≡ ∇² - (1/c²)∂²/∂t²)