To find the maximum value of the emf induced in the loop, we can use Faraday's law of electromagnetic induction. This law states that the induced emf in a closed loop is equal to the negative rate of change of the magnetic flux through the loop.

The formula for Faraday's law is:

E = -dΦ/dt

where E is the induced emf, and Φ is the magnetic flux.

The magnetic flux Φ through the loop is given by the product of the magnetic field B and the area A of the loop:

Φ = BA

Substituting B = B0sin(2πft) into the equation gives:

Φ = B0sin(2πft) * A

The area A of the loop is given by the formula for the area of a circle, πr^2, where r is the radius of the loop. Given that the diameter of the loop is 49.9 cm, the radius is 49.9/2 = 24.95 cm = 0.2495 m. So:

A = π(0.2495 m)^2 = 0.196 m^2

Substituting A = 0.196 m^2 into the equation for Φ gives:

Φ = B0sin(2πft) * 0.196 m^2

Now we can find the rate of change of Φ with respect to time t, dΦ/dt. The derivative of sin(2πft) with respect to t is 2πf cos(2πft). So:

dΦ/dt = B0 * 2πf * cos(2πft) * 0.196 m^2

The maximum value of cos(2πft) is 1, so the maximum value of dΦ/dt, and hence the maximum value of the induced emf E, is:

E = B0 * 2πf * 1 * 0.196 m^2

Substituting B0 = 5.1 nT = 5.1 * 10^-9 T and f = 109.8 MHz = 109.8 * 10^6 Hz into the equation gives:

E = 5.1 * 10^-9 T * 2π * 109.8 * 10^6 Hz * 0.196 m^2 = 0.207 mV

So the signal strength E going into the receiver is 0.207 mV.

Question

To find the maximum value of the emf induced in the loop, we can use Faraday's law of electromagnetic induction. This law states that the induced emf in a closed loop is equal to the negative rate of change of the magnetic flux through the loop.

The formula for Faraday's law is:

E = -dΦ/dt

where E is the induced emf, and Φ is the magnetic flux.

The magnetic flux Φ through the loop is given by the product of the magnetic field B and the area A of the loop:

Φ = BA

Substituting B = B0sin(2πft) into the equation gives:

Φ = B0sin(2πft) * A

The area A of the loop is given by the formula for the area of a circle, πr^2, where r is the radius of the loop. Given that the diameter of the loop is 49.9 cm, the radius is 49.9/2 = 24.95 cm = 0.2495 m. So:

A = π(0.2495 m)^2 = 0.196 m^2

Substituting A = 0.196 m^2 into the equation for Φ gives:

Φ = B0sin(2πft) * 0.196 m^2

Now we can find the rate of change of Φ with respect to time t, dΦ/dt. The derivative of sin(2πft) with respect to t is 2πf cos(2πft). So:

dΦ/dt = B0 * 2πf * cos(2πft) * 0.196 m^2

The maximum value of cos(2πft) is 1, so the maximum value of dΦ/dt, and hence the maximum value of the induced emf E, is:

E = B0 * 2πf * 1 * 0.196 m^2

Substituting B0 = 5.1 nT = 5.1 * 10^-9 T and f = 109.8 MHz = 109.8 * 10^6 Hz into the equation gives:

E = 5.1 * 10^-9 T * 2π * 109.8 * 10^6 Hz * 0.196 m^2 = 0.207 mV

So the signal strength E going into the receiver is 0.207 mV.

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