Problem Set 3

Problem Set 3

Due Friday, February 15, by 4pm in the homework locker

Problem 1 [10 points]
(a) Find the electric and magnetic fields, and the charge and current densities, corresponding to the potentials

V(r,t)=0 and A(r,t) = -1
4πε₀ qt
r² r̂

(b) Use the gauge function λ(r,t) = -(1/4πε₀)(qt/r) to transform the potentials, and comment on the result.
Problem 2 [10 points]
Suppose the electric and magnetic fields are:

E(r,t) = 1
4πε₀ q
r² θ(r-vt)e_r and B(r,t)=0

where e_r is the unit vector in the radial direction, and θ(x)=1 for x>0 and θ(x)=0 for x<0. Show that these fields satisfy all four of Maxwell's equations, and determine the charge density ρ and current density j. Describe the physical situation that gives rise to these fields.
Problem 3 [10 points]
Consider Maxwell's equations in the presence of charge and current sources ρ and j. Show that E and B now satisfy the inhomogeneous wave equation
∇²E − μ_oε_o∂²E/∂t² = g(r, t) and ∇²B − μ_oε_o∂²B/∂t² = h(r, t)
and express the functions g(r, t) and h(r, t) in terms of the charge density ρ(r, t) and the current density j(r, t) (i.e. the functions g and h should depend only on ρ and j and not on E and B).

Problem 4 [30 points total]

In lecture we defined the three dimensional Fourier transform of a function f(k) = FT[f(r)] by:

f(k) = 1
(2π)³ +∞
∫
−∞ d³r f(r) e^{−ik ⋅ r} and f(r) = +∞
∫
−∞ d³k f(k) e^{ik ⋅ r}

a) [5 points] If δ(r) is the three dimensionsl Dirac delta function, show that its Fourier transform is a constant independent of k

FT[δ(r)] = 1
(2π)³ so that δ(r) = 1
(2π)³ +∞
∫
−∞ d³k e^{ik ⋅ r}

b) [5 points] Using the result of part (a), show that the Fourier transform of the complex exponential e^ik_o⋅r is just a Dirac delta function in k-space, δ(k − k_o).

c) [5 points] Consider the Poisson equation for the electrostatic potential of a point charge q located at the origin, −∇²V(r) = q δ(r)/ε_o. By taking the Fourier transform of this equation, and using the results of the previous parts, find the Fourier transform V(k) of the solution to this problem V(r).

You already know that the solution to this problem is V(r) = q/(4πε_o|r|). Use this fact and your result above to show that the Fourier transform of 1/|r| is 4π/((2π)³ k²). This is an easier way to find the Fourier transform of 1/|r| than by doing the Fourier transform integration of 1/|r| directly!

d) [15 points] In lecture we discussed the decomposition of any vector function f(r) into its longitudinal (curlfree) and transverse (divergenceless) parts, f_L(r) and f_T(r), as given by Helmholtz's theorem:

f(r) = f_L(r) + f _T(r)

where

f_L(r) = − ∇ [ 1
4π
∫ d³r′ ∇′ ⋅ f(r′)
|r-r′|
] and f_T(r) = ∇x [ 1
4π
∫ d³r′ ∇′ x f(r′)
|r-r′|
]

Using the results above, find expressions for the Fourier transforms of the longitudinal and transverse parts of f(r). Your results should tell you why these parts have the names that they do!

Hint: Just substitute into the above formulas the expressions for the Fourier transforms of f(r) and 1/|r−r'| and rearrange the order of the integrations, do the integrals that you can, and then identify the Fourier coefficients of the longitudinal and transvese parts of f(r) from the result. You should then have found a relatively simple relation between f_L(k) and f_T(k) and the Fourier transform f(k) of f(r).