A uniform magnetic field pointing in the positive z direction fills a cylindrica

Question

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 (Figure 1) changes with time as B=Bmaxsin(?t).

A) Calculate the magnitude of the electric field that accompanies this changing magnetic field as a function of time tand radial distance r from the center of the magnetic field for r<R.

Express your answer in terms of all or some of the variables r, R, Bmax, ?, and t.

B) Calculate the magnitude of the electric field that accompanies this changing magnetic field as a function of time tand radial distance r from the center of the magnetic field for r>R

Explanation / Answer

Here ,

part a)

for the magnitude of electric field inside the area ,

for r < R

for the induced electric field ,

using Faraday's law

2pi * r * E = pi * r^2 * dB/dt

2pi * r * E = pi * r^2 d/dt(Bmax * sin(w * t))

E = r * Bmax * w/2 * cos(w * t)

the induced electric field is r * Bmax * w/2 * cos(w * t)

part B)

for the electric field outside the region

for the induced electric field ,

using Faraday's law

2pi * R * E = pi * r^2 * dB/dt

2pi * r * E = pi * r^2 d/dt(Bmax * sin(w * t))

E = r^2 * Bmax * w/(2R) * cos(w * t)

the induced electric field is r^2 * Bmax * w/(2R) * cos(w * t)

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