Question

Shrinking Loop. A circular loop of flexible iron wire has an initial circumference of 168 cm , but its circumference is decreasing at a constant rate of 15.0 cm/s due to a tangential pull on the wire. The loop is in a constant uniform magnetic field of magnitude 0.900 T , which is oriented perpendicular to the plane of the loop. Assume that you are facing the loop and that the magnetic field points into the loop.

Required:
Find the magnitude of the emf EMF induced in the loop after exactly time 8.00s has passed since the circumference of the loop started to decrease.

Answer 1

To find the magnitude of the emf induced in the loop after 8.00 seconds has passed since the circumference started to decrease, we can use Faraday's law of electromagnetic induction. We calculate the rate of change of magnetic flux through the loop based on the changing area of the loop, and then determine the magnitude of the emf induced in the loop. The emf induced is -253.30 V, indicating that the induced current flows in a direction that opposes the change in magnetic flux.

Step-by-step explanation:

To find the magnitude of the emf induced in the loop after 8.00 seconds has passed since the circumference started to decrease, we can use Faraday's law of electromagnetic induction. According to Faraday's law, the emf induced in a loop is equal to the rate of change of magnetic flux through the loop. In this case, as the loop shrinks, its area decreases, resulting in a decrease in magnetic flux.

We know that the circumference of the loop is decreasing at a constant rate of 15.0 cm/s. Using the formula for the circumference of a circle, we can determine the radius of the circle at the given time: r = C / (2*pi), where C is the circumference and pi is a mathematical constant approximately equal to 3.14159. Substituting the given values, we get r = 168 cm / (2*3.14159) = 26.79 cm.

Next, we can calculate the area of the loop as a function of time using the equation A = pi*r^2. Substituting the value of the radius at 8.00 seconds, we get A = 3.14159 * (26.79 cm)^2 = 2252.68 cm^2.

Since the magnetic field is perpendicular to the loop and uniform in magnitude, we can calculate the rate of change of magnetic flux as: dPhi/dt = B*dA/dt, where B is the magnitude of the magnetic field and dA/dt is the rate of change of the area.

Finally, we can calculate the magnitude of the emf induced in the loop as: EMF = -dPhi/dt. The negative sign indicates that the induced current flows in a direction that opposes the change in magnetic flux. Substituting the given values, we get EMF = -0.900 T * (2252.68 cm^2) / 8.00 s = -253.30 V.

Answer 2

103.1 V

Step-by-step explanation:

We are given that

Initial circumference=C=168 cm

`(dC)/(dt)=-15cm/s`

Magnetic field,B=0.9 T

We have to find the magnitude of the emf induced in the loop after exactly time 8 s has passed since the circumference of the loop started to decrease.

Magnetic flux=
`\phi=BA=B(\pi r^2)`

Circumference,C=
`2\pi r`

`r=(C)/(2\pi)`

`r=(168)/(2\pi)` cm

`(dr)/(dt)=(1)/(2\pi)(dC)/(dt)=(1)/(2\pi)(-15)=-(15)/(2\pi) cm/s`

`\int dr=-\int (15)/(2\pi)dt`

`r=-(15)/(2\pi)t+C`

When t=0

`r=(168)/(2\pi)`

`(168)/(2\pi)=C`

`r=-(15)/(2\pi)t+(168)/(2\pi)`

E=
`-(d\phi)/(dt)=-(d(B\pi r^2))/(dt)=-2\pi rB(dr)/(dt)`

`E=-2\pi(-(5)/(2\pi)t+(168)/(2\pi))B* -(15)/(2\pi)`

t=8 s

B=0.9

`E=2\pi* (15)/(2\pi)* 0.9(-(15)/(2\pi)(8)+(168)/(2\pi))`

`E=103.1 V`