# Magnetic Field due to Time Dependent Current

1. Apr 2, 2014

### Mysterious

1. The problem statement, all variables and given/known data
A long, straight, copper wire has a circular cross section with radius R, resistivity p and permittivity ε. If the current through the wire at any time t is sin(ωt) amperes, find the magnitude of the magnetic field B at time t a distance r from the centre of the wire for r > R.

2. Relevant equations
Ampere's Law:
μI = Bdl
Possibly Law of Biot-Savart:
B = μ/4π * (Idl x r)/r^2

3. The attempt at a solution
μI = Bdl
μ(sin(ωt))=B*dl (Since they are parallel)
μ(sin(ωt))=Bdl (Since B is constant radially around conductor)

This is where I reach a bottleneck. I don't know how to incorporate ε (since this is a magnetic field). I assume resistivity would be used in calculating the current I, but I don't know how that ties into the sinusoidal function.

2. Apr 6, 2014

### BruceW

maybe you don't need to use $\epsilon$ and $\rho$. Also, you have already used a certain kind of assumption about $\epsilon$ and $\rho$ to get your answer. So maybe the question is hoping that you will explicitly state this assumption about $\epsilon$ and $\rho$.

Your answer is essentially a kind of quasi-static approximation. (Not the most general answer for this question). But you have implicitly used the fact that copper is a good conductor, and assumed a certain relationship between $\omega$, $\epsilon$ and $\rho$. I'm not sure if that was your deliberate intention, or if you missed a few steps. But I think you have the answer they were looking for, but maybe without explaining under what approximation this answer will work.