Solution: Power Required for Equal Saturation Current
Key Concept: Saturation current $i$ is proportional to the number of photoelectrons emitted per second ($n_e$).
$$ n_e = \eta \times n_{photons} = \eta \times \frac{\text{Power}}{\text{Energy per photon}} $$
Step 1: Expression for Current
$$ i \propto \eta \left( \frac{P}{hc/\lambda} \right) \propto P \eta \lambda $$
$$ i \propto \eta \left( \frac{P}{hc/\lambda} \right) \propto P \eta \lambda $$
Step 2: Equate Currents
Given $i_1 = i_2$: $$ P_1 \eta_1 \lambda_1 = P_2 \eta_2 \lambda_2 $$ Let the unknown power of beam 2 be $W_2$ (and $W_1$ for beam 1 as per options). $$ W_1 \eta_1 \lambda_1 = W_2 \eta_2 \lambda_2 $$
Given $i_1 = i_2$: $$ P_1 \eta_1 \lambda_1 = P_2 \eta_2 \lambda_2 $$ Let the unknown power of beam 2 be $W_2$ (and $W_1$ for beam 1 as per options). $$ W_1 \eta_1 \lambda_1 = W_2 \eta_2 \lambda_2 $$
Step 3: Solve for $W_2$
$$ W_2 = W_1 \frac{\eta_1 \lambda_1}{\eta_2 \lambda_2} $$
$$ W_2 = W_1 \frac{\eta_1 \lambda_1}{\eta_2 \lambda_2} $$
Correct Option: (d) $W_1 \frac{\eta_1 \lambda_1}{\eta_2 \lambda_2}$