- #1
RedDeer44
- 3
- 1
- Homework Statement
- This is from Griffith , Introduction to Electrodynamics 4th Edition question 10.7
A time-dependent point charge q(t) at the origin, ##\rho(r, t) = q(t)\delta^3(r)##, is fed by a current ##J(r, t) = −\frac{1}{4π} \frac{\dot q}{r^2}) \hat r##
(a) Check that charge is conserved, by confirming that the continuity equation is obeyed.
(b) Find the scalar and vector potentials in the Coulomb gauge. If you get stuck, try working on (c) first.
(c) Find the fields, and check that they satisfy all of Maxwell’s equations.
- Relevant Equations
- $$A = \frac{\mu_0}{4\pi}\int \frac{ J}{|r-r'|}d^3r'$$
$$J = −\frac{1}{4π} \frac{\dot q}{r^2}) \hat r$$
I am having problem with part (b) finding the vector potential. More specifically when writing out the volume integral,
$$A = \frac{\mu_0}{4\pi r}\frac{dq}{dt}\int_{0}^{2\pi}\int_{0}^{\pi}\int_{0}^{?}\frac{1}{4\pi r'^2} r'^2sin\theta dr'd\theta d\phi$$
How do I integrate ##r'##?
The solution says because ##B## can only have ##r## component and only depend on ##r## and ##t## due to symmetry, and ##\nabla \cdot B = 0##, ##B = \nabla \times A = 0## also by using coulomb gauge we set ## \nabla\cdot A = 0##, ##A=0## or it could be any constant.
The argument makes sense to me, but I am not sure what is wrong with the integral, and how does it end up with 0.
$$A = \frac{\mu_0}{4\pi r}\frac{dq}{dt}\int_{0}^{2\pi}\int_{0}^{\pi}\int_{0}^{?}\frac{1}{4\pi r'^2} r'^2sin\theta dr'd\theta d\phi$$
How do I integrate ##r'##?
The solution says because ##B## can only have ##r## component and only depend on ##r## and ##t## due to symmetry, and ##\nabla \cdot B = 0##, ##B = \nabla \times A = 0## also by using coulomb gauge we set ## \nabla\cdot A = 0##, ##A=0## or it could be any constant.
The argument makes sense to me, but I am not sure what is wrong with the integral, and how does it end up with 0.