A rectangular loop with a=2.9 cm, and b=1.9 cm, moving with a velocity v=3.8 cm/s into an area where the magnetic field is uniform, pointing into the page, and whose magnitude is B= 450 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 0.04 seconds to do that, what is in volts the induced emf in the loop?
I know the answer to a is zero

To calculate the induced emf in the loop, we can use Faraday's law of electromagnetic induction, which states that the induced emf is equal to the rate of change of magnetic flux through the loop.

a) When the loop is moving in a region where the magnetic field is uniform, the induced emf can be calculated using the formula:

emf = -N * d(Φ) / dt

Where,
emf is the induced emf,
N is the number of turns in the loop,
d(Φ) / dt is the rate of change of magnetic flux through the loop.

In this case, we have a rectangular loop with dimensions a = 2.9 cm and b = 1.9 cm. The velocity of the loop is v = 3.8 cm/s.

The magnetic field strength is given as B = 450 mT (millitesla), which can be converted to tesla by dividing by 1000. So, B = 0.45 T.

The magnetic flux through the loop is given by the product of the magnetic field, the area of the loop, and the cosine of the angle between the magnetic field and the normal to the loop. Since the loop is being moved parallel to the magnetic field, the angle will be 0 degrees, and cos(0) = 1.

The area of the loop is given by the product of the sides a and b, so A = a * b.

Using the given values, we have:

A = 2.9 cm * 1.9 cm

Now, to find the rate of change of magnetic flux, we need to differentiate the area with respect to time:

dA / dt = d(a*b) / dt

Since both a and b are constants, their derivatives are zero, so the rate of change of area is zero.

Therefore, the induced emf in the loop when it is moving in a region where the magnetic field is uniform is zero.

b) When we pull two opposite ends of the loop apart until the overall area of the loop is zero, we are effectively reducing the area of the loop. This means that the rate of change of magnetic flux is no longer zero.

To calculate the induced emf in this case, we can use the same formula as before:

emf = -N * d(Φ) / dt

Since the area of the loop is decreasing, the rate of change of magnetic flux will be negative.

Now, given that it takes 0.04 seconds to reduce the area to zero, we can find the rate of change of magnetic flux by dividing the change in flux by the time interval:

d(Φ) / dt = Δ(Φ) / Δt

Since the overall area of the loop becomes zero, the change in flux is equal to the initial flux:

Δ(Φ) = Φ_initial

Using the formula for magnetic flux, Φ = B * A * cos(θ), we have:

Φ_initial = B * A_initial * cos(θ)

Now, the initial area of the loop is A_initial = 2.9 cm * 1.9 cm.

Substituting the values into the formula, we get:

Δ(Φ) / Δt = B * (2.9 cm * 1.9 cm) * cos(θ) / 0.04 s

Since the magnetic field is uniform and pointing into the page, the angle θ between the field and the normal to the loop will be 0 degrees, and cos(0) = 1.

So, Δ(Φ) / Δt = B * (2.9 cm * 1.9 cm) / 0.04 s

Substituting the given values, we have:

Δ(Φ) / Δt = (0.45 T) * (2.9 cm * 1.9 cm) / 0.04 s

And finally, using the formula for induced emf, we can calculate the value:

emf = -N * Δ(Φ) / Δt

Since the number of turns in the loop is not given, we cannot calculate the exact induced emf, but it will be a negative value.