Calculating the magnitude of the electric field

[Barry Melby]

Homework Statement

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.

Homework Equations

I'm not exactly sure which equations to use.

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?

 

[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.

 

[vela]

Homework Statement

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.

Homework Equations

I'm not exactly sure which equations to use.

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

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

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?