The figure shows the electric field inside a cylinder of radius R = 2.9 mm . The

Question

The figure shows the electric field inside a cylinder of radius R= 2.9 mm . The field strength is increasing with time as E= 1.5 ×108t2V/m , where t is in s . The electric field outside the cylinder is always zero, and the field inside the cylinder was zero for t<0 .

Part A

Find an expression for the electric flux e through the entire cylinder as a function of time.

Express your answer in terms of E and R and appropriate constants.

Part B

Find an expression for the magnetic field strength as a function of time at a distance r<R from the center.

Express your answer using two significant figures.

Part C

Evaluate the magnetic field strength at r= 2.0 mm , t= 2.2 s .

Express your answer to two significant figures and include the appropriate units.

Part D

Find an expression for magnetic field strength as a function of time at a distance r>R from the center.

Express your answer using two significant figures.

Part E

Evaluate the magnetic field strength at r= 4.5 mm , t= 2.5 s .

Express your answer to two significant figures and include the appropriate units.

Explanation / Answer

(a)

electric flux is = E * pi R^2 = 1.5 ×10^8 t2V/m * pi ( 2.9 * 10^-3)^2 = 3.96 * 10^3 t^2

(b)

closed integral B. ds = e_o u_o d ( flux)/ dt

(c)

B ( 2 pi r) =  e_o u_o d ( E) Pi r^2/ dt

B =  e_o u_o d ( E) Pi r^2/2 Pi r* dt

= r/ 2 c^2 ( 1.5 ×10^8 V/m) 2t

=1.66 * 10^-9 rt r <R

(d)

at r= 2.0 mm , t= 2.2 s .

B = 1.66 * 10^-9 ( 2.0 * 10^-3 m) ( 2.2 s) =7.304 * 10^-12 T

(e)

B ( 2 pi r) = e_o u_o d E/ dt * pi R^2 = R^2/ 2c^2 r ( 1.1 * 10^8 V/m) 2t) =1.027 * 10^-14 t/r

(f)

B = 1.027 * 10^-14 (2.5)/4.5 * 10^-3)

=5.71 * 10^-12 T

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