Question

Energy carried by an EM wave coming through the air can be used to light a bulb that is not connected to a battery or plugged into an electric outlet. Suppose a receiving antenna is attached to a bulb and the bulb is found to dissipate a maximum power of 1.05 W when the antenna is aligned with the electric field coming from a distant source. The wavelength of the source is large compared to the antenna length. When the antenna is rotated so it makes an angle of \(20.0^{\circ}\) with the incoming electric field, what is the power dissipated by the bulb?

Short answer

Answer: The power dissipated by the bulb when the antenna is rotated at an angle of \(20.0^{\circ}\) is approximately 0.979 W.

Step-by-step solution

Understanding the relationship with the angle and power dissipation

Since we know that the maximum power of 1.05 W is dissipated when the antenna is aligned with the electric field, the power will decrease when an angle is formed. The power dissipated will depend on the angle formed between the electric field and the antenna.

Using cosine function to relate the angle and power

The component of the electric field parallel to the antenna will be proportional to the cosine of the angle formed. The power dissipated by the bulb is proportional to the square of the electric field component along the antenna. Thus, we can write the power \(P\) at an angle of \(\theta\) as: \(P(\theta) = P_{max} \times \cos^2(\theta)\) Where \(P_{max}\) is the maximum power when the antenna is aligned with the electric field (1.05 W), and \(\theta\) is the angle formed.

Calculate the power at \(20.0^{\circ}\)

Now, we need to calculate the power \(P\) when the antenna is rotated by an angle of \(20.0^{\circ}\): \(P(20.0^{\circ}) = 1.05 \times \cos^2(20.0^{\circ})\) First, find the cosine of \(20.0^{\circ}\): \(\cos(20.0^{\circ}) \approx 0.9397\) Now, square the cosine and multiply it by the maximum power: \(P(20.0^{\circ}) = 1.05 \times (0.9397)^2 \approx 0.979\,\text{W}\) Therefore, the power dissipated by the bulb when the antenna is rotated at an angle of \(20.0^{\circ}\) is approximately 0.979 W.