Introduction:
In this scenario, we have a conducting metal circular wire loop placed perpendicular to a magnetic field that varies with time. We are required to determine the heat generated in the loop after a long time, when t tends to infinity.

Given Information:
- Radius of the circular wire loop: r
- Magnetic field: B = B′e^(-t/τ), where B′ and τ are constants
- Resistance of the loop: R

Explanation:
To find the heat generated in the loop, we need to calculate the power dissipated in the loop. The power can be obtained using the formula:
P = I^2 * R

Step 1: Find the current flowing through the loop
According to Faraday's law of electromagnetic induction, a changing magnetic field induces an electromotive force (EMF) in a closed loop of wire. The induced EMF is given by the equation:
EMF = -dΦ/dt

In this case, the magnetic field is changing with time, and since the loop is perpendicular to the field, the magnetic flux Φ through the loop is given by:
Φ = B * A = B * π * r^2

Differentiating Φ with respect to time, we get:
dΦ/dt = dB/dt * π * r^2

Using the given equation for the magnetic field B = B′e^(-t/τ), the rate of change of magnetic field dB/dt can be calculated as:
dB/dt = (-B′/τ) * e^(-t/τ)

Substituting this value into the equation for dΦ/dt, we have:
dΦ/dt = (-B′/τ) * e^(-t/τ) * π * r^2

This dΦ/dt represents the induced EMF in the loop. According to Ohm's law, the induced EMF is equal to the current flowing through the loop multiplied by the resistance:
EMF = I * R

Equating the above two equations, we can find the current flowing through the loop:
I = (-B′/τ) * e^(-t/τ) * π * r^2 / R

Step 2: Calculate the power dissipated in the loop
Now that we know the current flowing through the loop, we can calculate the power dissipated using the formula:
P = I^2 * R

Substituting the value of I, we get:
P = [(-B′/τ) * e^(-t/τ) * π * r^2 / R]^2 * R
P = (B′^2 / τ^2) * e^(-2t/τ) * π^2 * r^4

Step 3: Calculate the heat generated in the loop
To find the heat generated in the loop after a long time (t tends to infinity), we need to evaluate the power P as t approaches infinity:
lim(t→∞) P = lim(t→∞) [(B′^2 / τ^2) * e^(-2t/τ) * π^2 * r^4]

As t approaches infinity, the exponential term e^(-2t/τ) tends to zero. Therefore,