a rectangular loop with a=6.0 cm, and b=4.0 cm, moving with a velocity v=8.0 cm/s into an area where the magnetic field is uniform, pointing into the page, and whose magnitude is B= 440 mT.

a) What is in volts the induced emf in the loop when the loop is moving in a region where the magnetic field is uniform?

b) We stop the loop and instead, while keeping it at the region of uniform field, we pull two opposite ends of the loop apart until the overall area of the loop is zero. If it takes 0.21 seconds to do that, what is in volts the induced emf in the loop?

a) No change in the magnetic field and the area thus no flux change with time. Thus induced emf=0

b)Change in the area is=0.06*0.04 m^2-0= 0.0024 m^2
Thus the induced emf =change in flux
=B*(change in area)/t
=0.440*0.0024/0.21
=0.005 V

a) To find the induced emf in the loop while it is moving in a region with a uniform magnetic field, we can use Faraday's law of electromagnetic induction. According to Faraday's law, the induced emf is equal to the rate of change of magnetic flux through the loop.

The formula for calculating the induced emf is:

emf = -N * dΦ/dt

Where:
emf is the induced emf
N is the number of turns in the loop (since it is not provided, we assume it as 1)
dΦ/dt is the rate of change of magnetic flux through the loop.

The magnetic flux through the loop is given by the formula:

Φ = B * A * cosθ

Where:
Φ is the magnetic flux
B is the magnitude of the magnetic field (given as 440 mT or 0.44 T)
A is the area of the loop (given as a*b = 6.0 cm * 4.0 cm = 24 cm^2)
θ is the angle between the magnetic field and the normal to the loop, which is 0 degrees since the magnetic field is pointing into the page.

Plugging in the values into the formula for the flux, we get:

Φ = (0.44 T) * (24 cm^2) * (cos0°)
Φ = 10.56 T*cm^2

Now, we can find the rate of change of magnetic flux by taking the derivative of the flux with respect to time:

dΦ/dt = 0, as the magnetic field, area, and orientation are constant.

Finally, substituting the values into the formula for the induced emf, we get:

emf = -1 * (0 T*cm^2/s)
emf = 0 volts

Therefore, the induced emf in the loop is 0 volts when it is moving in a region with a uniform magnetic field.

b) To find the induced emf in the loop when we pull two opposite ends of the loop apart until the overall area becomes zero, we can again use Faraday's law of electromagnetic induction. However, in this case, the area of the loop is changing with time.

The formula for the induced emf remains the same:

emf = -N * dΦ/dt

To calculate dΦ/dt, we need to determine how the area of the loop is changing with time. We are given that it takes 0.21 seconds to reduce the overall area of the loop to zero.

We can find the rate of change of the area by dividing the change in area by the change in time:

dA/dt = (final area - initial area) / (final time - initial time)

The initial area of the loop is given as 24 cm^2 (as calculated in part a), and the final area is 0 cm^2. The initial time is 0 seconds, and the final time is 0.21 seconds.

Plugging in the values, we get:

dA/dt = (0 cm^2 - 24 cm^2) / (0.21 s - 0 s)
dA/dt = -24 cm^2 / 0.21 s
dA/dt = -114.3 cm^2/s

Now, substituting the value of dA/dt along with the other known values into the formula for the induced emf, we get:

emf = -1 * (-114.3 cm^2/s)
emf = 114.3 volts

Therefore, the induced emf in the loop is 114.3 volts when we pull two opposite ends of the loop apart until the overall area becomes zero.