Question

If two linear antennas having lengths in the ratio 2:3 are emitting radiations of wavelengths in the ratio 8:9, then the ratio of effective powers radiated by them are in the ratio:

• A. \( 9:16 \)
• B. \( 27:32 \)
• C. \( 16:27 \)
• D. \( 32:18 \)

Hint:

The power radiated by an antenna is proportional to \( L^2 / \lambda^2 \). Be sure to square the ratios of the lengths and wavelengths.

Answer 1

The Correct Option is A

Step 1:
The power radiated by an antenna is proportional to the square of the length and inversely proportional to the square of the wavelength. Thus, the ratio of the effective powers radiated is: \[ \frac{P_1}{P_2} = \left( \frac{L_1^2}{\lambda_1^2} \right) \Bigg/ \left( \frac{L_2^2}{\lambda_2^2} \right) \] Substitute the given ratios of lengths \( L_1/L_2 = 2/3 \) and wavelengths \( \lambda_1/\lambda_2 = 8/9 \): \[ \frac{P_1}{P_2} = \left( \frac{2^2}{3^2} \right) \Bigg/ \left( \frac{8^2}{9^2} \right) = \frac{4/9}{64/81} = \frac{4 \times 81}{9 \times 64} = \frac{9}{16} \] Thus, the ratio of the effective powers radiated is \( 9:16 \).