Question

A radio tower is transmitting \(30.0 \mathrm{~kW}\) of power equally in all directions. Assume that the radio waves that hit the Earth are reflected. a) What is the magnitude of the Poynting vector at a distance of \(12.0 \mathrm{~km}\) from the tower? b) What is the root-mean-square value of the electric force on an electron at this location?

Short answer

Answer: To find the magnitude of the Poynting vector, we first calculate the intensity, \(I\), and then find the electric field strength, \(E\). After finding these values, we can determine the root-mean-square value of the electric force on an electron, \(F_{rms}\). The magnitude of the Poynting vector and the root-mean-square value of the electric force at a distance of 12.0 km from the radio tower would require numerical solutions from the provided equations.

Step-by-step solution

Calculating the intensity of the transmitted power

Using the information given, we can find the intensity of the transmitted power at a distance of 12.0 km from the tower. Intensity, \(I\), is defined as power per unit area. We are given the power, \(P = 30.0~kW = 30.0 \times 10^3~W\), transmitted uniformly in all directions, so the area over which this power is transmitted is the surface of a sphere. The formula for the surface area of a sphere is \(A = 4\pi r^2\), where \(r\) is the radius of the sphere. We know the radius is 12.0 km or \(12.0 \times 10^3~m\): \(A = 4\pi (12.0 \times 10^3)^2\) To find the intensity, we divide the power by this area: \(I = \frac{30.0 \times 10^3}{4\pi (12.0 \times 10^3)^2}\)

Calculating the magnitude of the Poynting vector

The Poynting vector can be found by multiplying the intensity (power per unit area) by the impedance of free space, \(\eta_0\) which is approximately equal to \(377\Omega\). Poynting vector, \(S\): \(S = I\eta_0 = \frac{30.0 \times 10^3}{4\pi (12.0 \times 10^3)^2}\times 377\) We'll now calculate the magnitude of the Poynting vector, S.

Calculating the electric field strength

Now we'll calculate the electric field strength, \(E\), using the Poynting vector formula: \(S = \frac{1}{2}\eta_0 E^2\) Rearranging the equation to find \(E\): \(E = \sqrt{\frac{2S}{\eta_0}}\) Substitute the previously calculated value for S.

Calculating the root-mean-square value of the electric force

Now that we have the electric field strength, we can find the root-mean-square value of the electric force on an electron using the formula: \(F_{rms} = e \cdot E_{rms}\) where \(e\) is the elementary charge of an electron, approximately \(1.6 \times 10^{-19} C\), and \(E_{rms}\) is the root-mean-square value of the electric field strength, equal to \(\frac{E}{\sqrt{2}}\): \(F_{rms} = e \cdot \frac{E}{\sqrt{2}}\) Now, substitute the value of \(E\) calculated in Step 3 and find the root-mean-square value of the electric force on an electron, \(F_{rms}\).