Question

A point source of $100 \,W$ emits light with $5 \%$ efficiency At a distance of $5\, m$ from the source, the intensity produced by the electric field component is:

• A. $\frac{1}{40 \pi} \frac{W}{m^2}$
• B. $\frac{1}{10 \pi} \frac{W}{m^2}$
• C. $\frac{1}{20 \pi} \frac{W}{m^2}$
• D. $\frac{1}{2 \pi} \frac{W}{m^2}$

Answer 1

The Correct Option is A

To solve this problem, we need to calculate the intensity of light produced by the electric field component at a distance of 5 meters from a point source, given the power emitted by the source and its efficiency.

1. First, determine the power that is actually converted to light. The source emits light with an efficiency of \(5\%\). Therefore, the light power emitted is

\(P_{\text{light}} = 0.05 \times 100 \, \text{W} = 5 \, \text{W}\)

1. Next, calculate the intensity of light at a distance from a point source. The intensity \((I)\) of light from a point source is given by the formula:

\(I = \frac{P_{\text{light}}}{4 \pi r^2}\)

where \(r\) is the distance from the source. Substituting the given values,

\(I = \frac{5 \, \text{W}}{4 \pi (5 \, \text{m})^2}\)

This simplifies to

\(I = \frac{5}{4 \pi \times 25} \, \text{W/m}^2 = \frac{5}{100 \pi} \, \text{W/m}^2 = \frac{1}{20 \pi} \, \text{W/m}^2\)

1. Therefore, the calculated intensity is \(\frac{1}{20 \pi} \, \text{W/m}^2\). However, let's revisit the options given:

• \(\frac{1}{40 \pi} \, \text{W/m}^2\)
• \(\frac{1}{10 \pi} \, \text{W/m}^2\)
• \(\frac{1}{20 \pi} \, \text{W/m}^2\)
• \(\frac{1}{2 \pi} \, \text{W/m}^2\)

1. Notice that the calculated intensity matches one of the answer choices — specifically, \(\frac{1}{20 \pi} \, \text{W/m}^2\) is the correct calculation result but not matching the listed correct answer. Therefore, I believe there might be an error in the listed correct answer choice. The calculated \(\frac{1}{20 \pi} \, \text{W/m}^2\) aligns with one of the given answer options, indicating that this should be the intended correction.

The selected and calculated correct choice based on the logical assessment and step-by-step analysis is:

\(\frac{1}{20 \pi} \, \text{W/m}^2\)