Question

A uniform magnetic field is perpendicular to the plane of a
circular loop of diameter 7.6 cm formed from wire of diameter 3.2
mm and resistivity of 1.83 × 10^{-8}Ω·m. At what rate must
the magnitude of the magnetic field change to induce a 8.8 A
current in the loop?

Answer #1

Step 1:

Find the resistance of circular loop:

R = rho*L/A

rho = resistivity of material = 1.83*10^-8 ohm-m

A = Cross-sectional area of wire = pi*d^2/4

d = diameter of wire = 3.2 mm = 3.2*10^-3 m

L = length of wire = circumference of loop = pi*D

D = diameter of circular loop = 7.6 cm = 7.6*10^-2 m

So,

R = 1.83*10^-8*pi*7.6*10^-2/(pi*(3.2*10^-3)^2/4)

R = 5.43*10^-4 ohm

Step 2:

Now Induced EMF in loop will be:

EMF = I*R

EMF = 8.8*5.43*10^-4

EMF = 4.78*10^-3 V

Step 3:

Induced EMF in circular loop in uniform magnetic field is given by:

EMF = N*d(phi)/dt

phi = magnetic flux = B*A1

EMF = N*d(B*A)/dt = N*A1*|dB/dt|

N = number of loops = 1

A1 = Area of circular loop = pi*D^2/4

|dB/dt| = magnitude of rate of change in magnetic field

|dB/dt| = EMF/A = 4*EMF/(N*pi*D^2)

|dB/dt| = 4*4.78*10^-3/(1*pi*(7.6*10^-2)^2)

**|dB/dt| = 1.05 T/sec**

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