Question

The magnetic field inside a 4.0-cm-diameter superconducting solenoid varies sinusoidally between 8.0 T and 12.0 T at a frequency of 7.0Hz .What is the maximum electric field strength at a point 2.0cm from the solenoid axis? What is the value of B at the instant E reaches its maximum value?

Answer 1

Step 1: Using Faraday's law, find electric field inside the solenoid:

We know that:

= Magnetic flux = B.A

So, Inside the solenoid for r < R,

So,

E.(2*pi*r) = -pi*r^2*(dB/dt)

E = (-r/2)*(dB/dt)

Now given that magnetic field varies sinusoidally between 8.0 and 12.0 T, with frequency of 7.0 Hz, So

B(t) = B0 + B1*sin (w*t)

w = 2*pi*f = 2*pi*7.0 = 14.0*pi

Since we know that

-1   sin (14.0*pi*t) 1

-2 2.0*sin (14.0*pi*t) 2

10 - 2 10 + 2.0*sin (14.0*pi*t) 10 + 2

8 10 + 2.0*sin (14.0*pi*t) 12

So,

8.0 B(t) 12.0

B(t) = 10.0 + 2.0*sin (14.0*pi*t)

dB/dt = 0 + 2.0*14.0*pi*cos (14.0*pi*t) = 28.0*pi*cos (14.0*pi*t)

Now electric field is given by:

E = -(r/2)*(dB/dt) = -(r/2)*28.0*pi*cos (14.0*pi*t)

E = -14.0*pi*r*cos (14.0*pi*t)

At r = 2.0 cm from the solenoid axis when electric field is maximum then cos (14.0*pi*t) = -1, So

Here r = 2.0 cm = 0.02 m

So,

E_max = -14.0*pi*0.02*(-1)

E_max = 0.8796 V/m

E_max = 0.88 V/m

Part B.

At the instant when electric field is maximum than since cosine function has value of (-1), So at these time sine function will have value of 0

(Remember at pi angle, since cos pi = -1, So at the same point sin pi = 0)

So when sine function is zero, then

B(t) = 10.0 + 2.0*sin (14.0*pi*t)

B(t) = 10.0 + 2.0*0 = 10.0 T

B(t) = 10.0 T

Let me know if you've any query.