A loop of wire in the shape of a rectangle of width w and length L a current l l

Question

A loop of wire in the shape of a rectangle of width w and length L a current l lie on a tabletop as shown in the figure below. Determine the magnetic flux through the loop due to the current l. (Use any variable stated above along with the following as necessary: mu_0 and pi) Suppose the current is changing with time according to l = a+ bt, where o and b are constants. Determine the magnitude of the emf that is induced in the loop if a - 2 A, b = -9 A/s, h =1.00 cm, w = 10 cm, and L = 1.00 m. What is the direction of the induced current in the rectangle?

Explanation / Answer

(a)
Using Ampere’s law,
Magnetic field due to a current-carrying wire at a distance r away is, B = (uo*I)/(2**r)
Net Flux is given by,
= B.dA
= (uo*I*L)/(2*) dr/r
= (uo*I*L)/(2*) * ln((h+w)/h)

(b)
I = a + b*t

e = -d/dt
e = -d/dt (uo*I*L)/(2*) * ln((h+w)/h)
e = - (uo*L)/(2*) *ln((h+w)/h) * dI/dt

dI/dt = b
So,
e = - (uo*L*b)/(2*) *ln((h+w)/h)

Substituing Values,
e = - (2*10^-7*1.0*(-9)) * ln(11)
e = 4.32 * 10^-6 V

(c)
The straight wire carrying a current I produces a magnetic flux into the page through the rectangular loop. By Lenz’s law, the induced current in the loop must be flowing counterclockwise in order to produce a magnetic field out of the page to counteract the increase in inward flux.

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