How Is EMF Induced in a Shrinking Square Loop?

[supersunshine]

Homework Statement

"A square loop of wire is held in a uniform magnetic field of .24 T directed perpendicular to the plane of the loop. The length of each side of the square is decreasing at a constant ratee of 5.0 cm/s. What is the emf induced in the loop when the length 12 cm?

Homework Equations

EMF= -d(flux)/dt = B . d(area)

The Attempt at a Solution

I think once the rate of change for the area is determined you can multiply it by the magnetic field. however, I am confused about determining the change in flux from the given variables. Is there going to be flux on each side of the square or only on certain sides? for dA can you mu ltiply the rate of change of the side by the length of a side and square that value?

thanks

 

[nrqed]

Homework Statement

"A square loop of wire is held in a uniform magnetic field of .24 T directed perpendicular to the plane of the loop. The length of each side of the square is decreasing at a constant ratee of 5.0 cm/s. What is the emf induced in the loop when the length 12 cm?

Homework Equations

EMF= -d(flux)/dt = B . d(area)

The Attempt at a Solution

I think once the rate of change for the area is determined you can multiply it by the magnetic field. however, I am confused about determining the change in flux from the given variables. Is there going to be flux on each side of the square or only on certain sides?

I am not sure what you mean by this. You care only about the flux through the surface enclosed by th eloop

for dA can you mu ltiply the rate of change of the side by the length of a side and square that value?

thanks

$A = L^2$. So what is $\frac{dA}{dt}$?

 

[m2003]

I would like to clarify some concepts and provide a step-by-step solution to this problem.

First, let's define some terms. EMF stands for electromotive force, which is the force that causes a current to flow in a circuit. In this case, it is induced by a changing magnetic field. Flux is a measure of the magnetic field passing through a given area.

To solve this problem, we can use Faraday's law, which states that the induced EMF is equal to the negative rate of change of magnetic flux through the loop. In other words, EMF = -d(flux)/dt.

In this scenario, the magnetic field is constant at 0.24 T, and the loop is decreasing in size at a rate of 5.0 cm/s. We can use the equation for the area of a square (A = s^2, where s is the length of a side) to determine the rate of change of the area. Since all sides of the square are decreasing at the same rate, we can use any side length in our calculations. Let's use the given length of 12 cm.

At a length of 12 cm, the area of the square is 12 cm x 12 cm = 144 cm^2. Taking the derivative with respect to time, we get dA/dt = 2s(ds/dt) = 2(12 cm)(-5.0 cm/s) = -120 cm^2/s.

Now, we can plug this value into the equation for EMF: EMF = -d(flux)/dt = (-0.24 T)(-120 cm^2/s) = 28.8 V.

To answer your question about the flux on each side of the square, yes, there will be flux passing through each side. However, since the magnetic field is perpendicular to the plane of the loop, the flux through each side will be the same. Therefore, we only need to consider the flux through one side.

In summary, the EMF induced in the square loop is 28.8 V when the length of each side is 12 cm. I hope this explanation helps clarify the concepts and the solution process.