`10^-3` W of `5000A` light is directed on a photoelectric cell. If the current in the cell is `0.16muA`, the percentage of incident photons which produce photoelectrons, is

Options

A. `40%`
B. `0.04%`
C. `20%`
D. `10%`

Explanation

To solve the problem step by step, we will calculate the percentage of incident photons that produce photoelectrons in a photoelectric cell.

### Step 1: Calculate the number of electrons emitted per unit time
Given:
- Current (I) = 0.16 µA = \(0.16 \times 10^{-6}\) A
- Charge of an electron (e) = \(1.6 \times 10^{-19}\) C

Using the formula for current:
\[ I = \frac{Q}{T} \implies Q = I \times T \]
The number of electrons emitted per unit time (n) can be calculated as:
\[ n = \frac{I}{e} \]
Substituting the values:
\[ n = \frac{0.16 \times 10^{-6}}{1.6 \times 10^{-19}} = 10^{12} \text{ electrons per second} \]

### Step 2: Calculate the energy of each photon
Given:
- Wavelength (λ) = 5000 Å = \(5000 \times 10^{-10}\) m
- Planck's constant (h) = \(6.63 \times 10^{-34}\) J·s
- Speed of light (c) = \(3 \times 10^{8}\) m/s

The energy (E) of each photon can be calculated using the formula:
\[ E = \frac{hc}{\lambda} \]
Substituting the values:
\[ E = \frac{(6.63 \times 10^{-34})(3 \times 10^{8})}{5000 \times 10^{-10}} = \frac{1.989 \times 10^{-25}}{5 \times 10^{-7}} = 3.978 \times 10^{-19} \text{ J} \]

### Step 3: Calculate the number of photons incident per unit time
Given:
- Power (P) = \(10^{-3}\) W

The number of photons (N) incident per unit time can be calculated using:
\[ N = \frac{P}{E} \]
Substituting the values:
\[ N = \frac{10^{-3}}{3.978 \times 10^{-19}} \approx 2.51 \times 10^{16} \text{ photons per second} \]

### Step 4: Calculate the percentage of incident photons that produce photoelectrons
The percentage of incident photons that produce photoelectrons can be calculated as:
\[ \text{Percentage} = \left(\frac{n}{N}\right) \times 100 \]
Substituting the values:
\[ \text{Percentage} = \left(\frac{10^{12}}{2.51 \times 10^{16}}\right) \times 100 \approx 0.0398\% \]

### Step 5: Final Result
The percentage of incident photons which produce photoelectrons is approximately:
\[ \text{Percentage} \approx 0.04\% \]