# Calculating the magnitude of the electric field

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1. Oct 22, 2015

### Barry Melby

1. The problem statement, all variables and given/known data
A uniform magnetic field pointing in the positive z-direction fills a cylindrical volume of space of radius R whose central axis is the z axis. Outside this region, there is no magnetic field. The magnitude of the magnetic field in changes with time as B = Bmax sin(ωt).

a. Calculate the magnitude of the electric field that accompanies this changing magnetic field as a function of time t and radial distance rfrom the center of the magnetic field for r < R.

b. Calculate the magnitude of the electric field that accompanies this changing magnetic field as a function of time t and radial distance rfrom the center of the magnetic field for r>R.

2. Relevant equations
I'm not exactly sure which equations to use.

3. The attempt at a solution

a)

E(2*pi*r) = pi*r^2 (dB/dt)
E = r/2 (dB/dt)
E = [omega*r*Bmax(sin(omega*t))]/2

This however is incorrect.

b)

E(2*pi*r) = pi*R^2 (dB/dt)
E = R^2/2r (dB/dt)
E = [omega*R^2*Bmax (cos(omega*t))]/2r

Which also appears to be incorrect. Where did I go wrong?

2. Oct 22, 2015

### andrewkirk

The Maxwell-Faraday equation describes the electric fields arising from time-varying magnetic fields. It easily gives you the curl of E and the EMF around a loop at the given radius. I forget how one extracts the E vector from that but no doubt it is straightforward.

3. Oct 25, 2015

### vela

Staff Emeritus
Probably just a typo, but the sine should be a cosine. You're also missing a negative sign (Lenz's law).