A circular loop of wire can be used as a radio antenna. If an 18.0-cm-diameter antenna i...

A circular loop of wire can be used as a radio antenna. If an 18.0-cm-diameter antenna is located 2.50 km from a 95.0-MHz source with a total power of 55.0 kW, what is the maximum emf induced in the loop? The orientation of the antenna loop and the polarization of the wave are as shown in Fig. 32.24. Assume that the source radiates uniformly in all directions.

Solution Guide

Identify and Set Up

1. The plane of the antenna loop is perpendicular to the direction of the wave’s oscillating magnetic field. This causes a magnetic flux through the loop that varies sinusoidally with time. By Faraday’s law, this produces an emf equal in magnitude to the rate of change of the flux. The target variable is the magnitude of this emf.

2. Select the equations that you will need to find (i) the intensity of the wave at the position of the loop, a distance r = 2.50 km from the source of power P = 55.0 kW; (ii) the amplitude of the sinusoidally varying magnetic field at that position; (iii) the magnetic flux through the loop as a function of time; and (iv) the emf produced by the flux.

Execute

3. Find the wave intensity at the position of the loop.

4. Use your result from step 3 to write expressions for the time-dependent magnetic field at this position and the time-dependent magnetic flux through the loop.

5. Use the results of step 4 to find the time-dependent induced emf in the loop. The amplitude of this emf is your target variable.

Evaluate

6. Is the induced emf large enough to detect? (If it is, a receiver connected to this antenna will pick up signals from the source.)