A circular loop of wire with radius 0.0450 m and resistance 0.154 R is in a region of spatially uniform magnetic. The magnetic field is directed out of the plane. The magnetic field has an initial value of 7.89 T and is decreasing at a rate of -0.630 T/s.

What is the rate at which electrical energy is being dissipated by the resistance of the loop?

I know that the rate is equal to P = I^2R

So I have to find the induced emf to find the current I.

I tried to find the induced emf using Faraday's Law and got 4.01*10^-3 V. But I don't know what to do with the initial value of the magnetic field 7.89 T.

To find the rate at which electrical energy is being dissipated by the resistance of the loop, you need to calculate the current passing through the loop.

You correctly noted that the rate of dissipated power is equal to P = I^2R, where R is the resistance of the loop. To determine the current, we need to calculate the induced electromotive force (emf) in the loop, which can be found using Faraday's Law.

Faraday's Law states that the emf induced in a coil equals the negative rate of change of the magnetic flux through the coil. Mathematically, this can be written as:

ε = -dΦ/dt

Where ε is the induced emf, Φ is the magnetic flux, and dt is the change in time. In this case, we have a circular loop with a spatially uniform magnetic field.

The magnetic flux through a loop is given by the product of the magnetic field (B) and the area (A) of the loop. Since the magnetic field is directed out of the plane and the loop is circular, we can write:

Φ = BA = πr^2B

Where r is the radius of the loop.

Now, to calculate the induced emf, we take the derivative of the above equation with respect to time:

dΦ/dt = πr^2(dB/dt)

Substituting the values given in the problem, we have:

dΦ/dt = π(0.0450 m)^2(-0.630 T/s)

Evaluating this expression will give you the rate of change of magnetic flux through the loop, which is equal to the induced emf (ε).

ε = -π(0.0450 m)^2(0.630 T/s)

Next, you can use Ohm's Law to calculate the current flowing through the loop. Ohm's Law states that the current (I) is equal to the emf (ε) divided by the resistance (R). In this case:

I = ε/R

Substituting the values given in the problem, you can calculate the current (I).

Finally, you can substitute the value of the current (I) back into the power equation (P = I^2R) to find the rate at which electrical energy is being dissipated by the resistance of the loop.