A rectangular conducting loop with dimensions w-24.0 cm and h 84.0 cm is placed

Question

A rectangular conducting loop with dimensions w-24.0 cm and h 84.0 cm is placed a distance a=5.00 cm from a long, straight wire carrying current 1-7.00 A in the downward direction (see figure below) (a) What is the magnitude of the magnetic flux through the loop? (b) If the current in the wire is increased linearly from 7.00 A to 16.0 A in 0.230 s, what is the magnitude of the induced emf in the loop? (c) What is the direction of the current that is induced in the loop during this time interval? O clockwise O counterclockwise

Explanation / Answer

w = 24 cm = 0.24 m

h = 84 cm = 0.84 m

a = 0.05 m = 5 cm

I = 7 A

a. at distance a < x < a + w

magnetic field due to the wire B(x) = 2kI/x

magnetic flux = df = 2kIh*dx/x

total flux = integrate df from x = a to x = a + w

f = 2kIh*ln(1 + w/a)

f = 2*10^-7*7*0.84*ln(1 + 0.24/0.05) = 2.06724*10^-6 T m^2

b. dI/dt = (16 - 7)/0.23 = 39.13 A/s

then

df/dt = 2kh*ln(1 + w/a)*dI/dt = 1.15558*10^-5 V

c. the direction of current is counterclockwise

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