A circular loop of flexible iron wire has an initial circumference of 168cm , but its circumference is decreasing at a constant rate of 15.0cm/s due to a tangential pull on the wire. The loop is in a constant uniform magnetic field of magnitude 1.00T , 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.

Find the magnitude of the EMF induced in the loop after exactly time 9.00s has passed since the circumference of the loop started to decrease.

To solve this problem, we can use Faraday's law of electromagnetic induction, which states that the electromotive force (EMF) induced in a loop of wire is equal to the rate of change of the magnetic flux through the loop.

The magnetic flux through a loop is given by the equation:

Φ = B * A

where Φ is the magnetic flux, B is the magnetic field, and A is the area of the loop.

In this case, the loop is circular, so the area can be calculated using the equation:

A = π * r^2

where r is the radius of the loop.

Since the circumference of the loop is given as 168 cm, we can find the radius using the equation:

C = 2 * π * r

168 cm = 2 * π * r
r = 168 cm / (2 * π) = 26.8 cm

Now, since the circumference of the loop is decreasing at a rate of 15.0 cm/s, the radius is also decreasing at the same rate. So at time t seconds, the radius can be given by the equation:

r(t) = 26.8 cm - (15.0 cm/s) * t

Now, we can calculate the area of the loop at time t:

A(t) = π * [r(t)]^2

Next, we need to calculate the rate of change of the magnetic flux. We can differentiate the equation for the magnetic flux with respect to time:

dΦ/dt = dB/dt * A + B * dA/dt

But since the magnetic field is constant (1.00 T) and not changing with time, the rate of change of the magnetic field is zero (dB/dt = 0). Therefore, the equation simplifies to:

dΦ/dt = B * dA/dt

Since the radius is decreasing at a constant rate of 15.0 cm/s, we can find the rate of change of the area as:

dA/dt = 2 * π * r * dr/dt
= 2 * π * (26.8 cm - 15.0 cm/s * t) * (-15.0 cm/s)

Now, we can substitute the values into the equation for the rate of change of the magnetic flux:

dΦ/dt = (1.00 T) * [2 * π * (26.8 cm - 15.0 cm/s * t) * (-15.0 cm/s)]

Finally, we can find the magnitude of the induced EMF using Faraday's law:

EMF = -dΦ/dt

EMF = -(1.00 T) * [2 * π * (26.8 cm - 15.0 cm/s * 9.00 s) * (-15.0 cm/s)]

EMF = 1.00 T * [2 * π * (26.8 cm - 135 cm) * 15.0 cm/s]

EMF = 1.00 T * [2 * π * (-108.2 cm) * 15.0 cm/s]

EMF = 1.00 T * [-2 * π * 108.2 cm * 15.0 cm/s]

EMF ≈ -1018 V (rounded to 3 significant figures)

Therefore, the magnitude of the EMF induced in the loop after exactly 9.00 seconds has passed since the circumference of the loop started to decrease is approximately 1018 volts.

To find the magnitude of the EMF induced in the loop, we can use Faraday's Law of electromagnetic induction. According to Faraday's Law, the EMF induced in a circuit is equal to the rate of change of magnetic flux through the circuit.

The magnetic flux through the loop can be calculated using the formula:
Φ = BA

Where Φ is the magnetic flux, B is the magnetic field strength, and A is the area of the loop.

The area of the loop can be calculated using the formula:
A = πr^2

Where A is the area, and r is the radius of the loop.

Initially, the circumference of the loop is 168 cm. We can find the radius by dividing the circumference by 2π:
r = C / (2π)
r = 168 cm / (2π)

The rate of change of the circumference is given as 15.0 cm/s. So, the rate of change of the radius can be calculated as:
dr/dt = -15.0 cm/s / (2π)

Now, we can differentiate the area of the loop with respect to time to find the rate of change of the area:
dA/dt = 2πr * dr/dt

Substituting the value of dr/dt, we can find dA/dt.

Next, we can substitute the values of B (1.00 T) and dA/dt into the formula for the rate of change of magnetic flux:
dΦ/dt = B * dA/dt

This gives us the rate of change of magnetic flux through the loop.

Finally, we can multiply the rate of change of magnetic flux by the time (9.00 s) to find the magnitude of the EMF induced in the loop:
EMF = dΦ/dt * t

By substituting the known values into the above equation, we can calculate the magnitude of the EMF induced in the loop after exactly 9.00 seconds.

L2=L1- (ΔL/Δt) •t=168-135=33 cm

EMF=-dΦ/dt =- d(B•A•cosα)/ dt.
cosα=1.
L1=2•π•R1
R1=L1/2•π=168/2•3.14=26.7 cm
A1=π•R1²=3.14•26.7²=2246 cm²=0.2246 m²

L2=2•π•R2
R2=L2/2•π=33/2•3.14=2.25 cm
A2=π•R2²=3.14•2.25²=86.7 cm²=0.0087m²
EMF=-dΦ/dt =- d(B•A•cosα)/dt.
cosα=1.
EMF=-B•ΔA/Δt=
= -1•(0.2246-0.0087)/9=0.024 V