Question

Public television station KQED in San Francisco broadcasts a sinusoidal radio signal at a power of 316 kW. Assume that the wave spreads out uniformly into a hemisphere above the ground.A. At a home 6.00km away from the antenna, what average pressure does this wave exert on a totally reflecting surface?B. At a home 6.00km away from the antenna, what are the amplitudes of the electric and magnetic fields of the wave?C. At a home 6.00km away from the antenna, what is the average density of the energy this wave carries?D. For the energy density in part (c), what percentage is due to the electric field?E. For the energy density in part (c), what percentage is due to the magnetic field?

Answer 1

To find the average pressure exerted by the wave on a reflecting surface 6.00 km away from the antenna, calculate the intensity of the wave. To find the amplitudes of the electric and magnetic fields, use the intensity and the formula for average power flow in a wave. To find the average density of the energy carried by the wave, use the intensity and the formula for energy density of an electromagnetic wave. To find the percentages of the energy density due to the electric and magnetic fields, use the formulas for energy density and the amplitudes of the fields.

Step-by-step explanation:

A. To find the average pressure exerted on a totally reflecting surface 6.00 km away from the antenna, we need to calculate the intensity of the wave at that distance. The intensity is given by the power divided by the area. Since the power is uniformly spread over a hemisphere, we can consider the area of the hemisphere as the surface area of a sphere with radius 6.00 km. Therefore, the intensity is given by:

1. Calculate the surface area of the sphere using the formula A = 4πr².
2. Divide the power by the surface area to obtain the intensity.

B. To find the amplitudes of the electric and magnetic fields, we can use the intensity of the wave and the formula for the average power flow in a wave. The average power flow is given by:

1. Calculate the average power flow using the formula P = ½ε₀cE₀².
2. Rearrange the formula to solve for E₀.

C. To find the average density of the energy carried by the wave, we can use the intensity and the formula for the energy density of an electromagnetic wave. The energy density is given by:

1. Calculate the energy density using the formula u = ε₀cE₀².
2. Divide the energy density by the volume of the hemisphere (which is equivalent to the volume of a sphere with radius 6.00 km) to obtain the average density of the energy.

D. To find the percentage of the energy density due to the electric field, we can use the formula for the energy density of an electromagnetic wave and the amplitudes of the electric and magnetic fields. The energy density is given by:

1. Calculate the energy density using the formula u = ½ε₀cE₀² + ½B₀²/μ₀.
2. Divide the energy density due to the electric field by the total energy density and multiply by 100 to obtain the percentage.

E. To find the percentage of the energy density due to the magnetic field, we can use the formula for the energy density of an electromagnetic wave and the amplitudes of the electric and magnetic fields. The energy density is given by:

1. Calculate the energy density using the formula u = ½ε₀cE₀² + ½B₀²/μ₀.
2. Divide the energy density due to the magnetic field by the total energy density and multiply by 100 to obtain the percentage.