in an experimental set up to study the photoelectric effect a point soure fo light of power `3.2xx10^(-3)` W was taken. The source can emit monoenergetic photons of energy 5eV and is located at a distance of 0.8 m from the centre of a stationary metallic sphere of work-function 3.0 eV. The radius of the sphere is `r = 8..10^(-3)` m. The efficiency of photoelectric emission is one for every `10^6` incident photons. Based on the information given above answer the questions given below. (Assume that the sphere is isolated and photoelectrons are instantly swepts away after the emission).
It was observed that after some time emission of photoelectrons from the sphere stopped. Charge on the sphere when the photon emission stops is

Options

A. `16piepsilon_0^r` coulomb
B. `8piepsilon_0^r` coulomb
C. `15piepsilon_0^r` coulomb
D. `20piepsilon_0^r` coulomb

Explanation

To find the charge on the sphere when the emission of photoelectrons stops, we can follow these steps:

### Step 1: Calculate the Stopping Potential
The stopping potential (V₀) can be calculated using the formula:
\[ V₀ = \frac{E_{incident} - W}{e} \]
where:
- \( E_{incident} \) = energy of the incident photon = 5 eV
- \( W \) = work function of the material = 3 eV

Substituting the values:
\[ V₀ = 5 \, \text{eV} - 3 \, \text{eV} = 2 \, \text{eV} \]

### Step 2: Relate Stopping Potential to Charge
The stopping potential is also related to the charge (q) on the sphere and its radius (r) by the equation:
\[ V₀ = \frac{q}{4 \pi \epsilon_0 r} \]
where:
- \( \epsilon_0 \) = permittivity of free space

### Step 3: Rearranging the Equation for Charge
Rearranging the equation to solve for charge (q):
\[ q = V₀ \cdot 4 \pi \epsilon_0 r \]

### Step 4: Substitute the Values
Now substitute the values into the equation. We know:
- \( V₀ = 2 \, \text{V} \)
- \( r = 8 \times 10^{-3} \, \text{m} \)
- \( \epsilon_0 \approx 8.85 \times 10^{-12} \, \text{C}^2/\text{N m}^2 \)

Calculating q:
\[ q = 2 \, \text{V} \cdot 4 \pi (8.85 \times 10^{-12} \, \text{C}^2/\text{N m}^2) (8 \times 10^{-3} \, \text{m}) \]

### Step 5: Perform the Calculation
Calculating the numerical value:
\[ q = 2 \cdot 4 \cdot 3.14 \cdot (8.85 \times 10^{-12}) \cdot (8 \times 10^{-3}) \]
\[ q \approx 2 \cdot 12.56 \cdot 8.85 \times 10^{-12} \cdot 8 \times 10^{-3} \]
\[ q \approx 2.24 \times 10^{-13} \, \text{C} \]

### Conclusion
The charge on the sphere when the emission of photoelectrons stops is approximately:
\[ q \approx 2.24 \times 10^{-13} \, \text{C} \]

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