A spatially uniform magnetic field B changes with time and induces an electric f

Question

A spatially uniform magnetic field B changes with time and induces an electric field E in a region of three-dimensional space containing the origin. The electric field is E = k(-sin theta, cos theta) N/C at the point (x, y, z), where theta = arctan(y/x), and k = 5.3. (a) What is the rate of change of the magnetic flux through the circle x 2 + y 2 = 4.0 m^2 in the xy plane? Assume that the area normal points in the positive z direction. (b) Assuming that the flux is zero at time t = 0 s, what is the flux through the circle when t = 10 s? (c) What is the magnetic field strength at t = 10 s?

Explanation / Answer

A)close loop integral E(vector).dl(vector) =d/dt integral(B.dA)

close loop integral E.(-rsin(theta)i+rcos(theta)j)d(theta) = pi*r^2*dB/dt

r=2,k=5.3

integral (from theta=0 to theta= 2pi) (-ksin(theta) i + kcos(theta)j).(-rsin(theta)i+rcos(theta)j)d(theta) =66.6

dB/dt=66.6/(pi*r^2)=5.3

B)

B=5.3t

B=53 at t=10

so flux=B.Pi*r^2 =666.01

C)

B=53

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