A copper ball of radius 1 cm and work function 4.47eV is irradiated with ultraviolet radiation of wavelength 2500 Å. The effect of irradiation results in the emission of electrons from the ball, Further the ball will acquire charge and due to this there will be a finite value of the potential on the ball. The charge acquired by the ball is :

Options

A. `5.5 xx 10^(-13) C`
B. `7.5 xx 10^(-13) C`
C. `4.5 xx 10^(-12) C`
D. `2.5 xx 10^(-11) C `

Explanation

To solve the problem step by step, we will follow the concepts of the photoelectric effect and electrostatics.

### Step 1: Calculate the energy of the incident photon
The energy of the photon can be calculated using the formula:
\[ E = \frac{hc}{\lambda} \]
where:
- \( h = 6.626 \times 10^{-34} \, \text{Js} \) (Planck's constant)
- \( c = 3 \times 10^8 \, \text{m/s} \) (speed of light)
- \( \lambda = 2500 \, \text{Å} = 2500 \times 10^{-10} \, \text{m} \) (wavelength in meters)

Substituting the values:
\[ E = \frac{(6.626 \times 10^{-34})(3 \times 10^8)}{2500 \times 10^{-10}} \]

### Step 2: Convert the energy from Joules to electron volts
Since 1 eV = \( 1.6 \times 10^{-19} \, \text{J} \), we can convert the energy calculated in Joules to electron volts by dividing by \( 1.6 \times 10^{-19} \).

### Step 3: Determine the kinetic energy of the emitted electrons
Using the photoelectric equation:
\[ K.E. = E - \phi \]
where \( \phi = 4.47 \, \text{eV} \) (work function of copper).

### Step 4: Calculate the potential developed on the ball
The potential \( V \) can be calculated using the kinetic energy:
\[ K.E. = eV \]
where \( e \) is the charge of an electron (\( 1.6 \times 10^{-19} \, \text{C} \)). Rearranging gives:
\[ V = \frac{K.E.}{e} \]

### Step 5: Calculate the charge acquired by the ball
Using the formula for potential due to a point charge:
\[ V = \frac{kQ}{R} \]
where:
- \( k = 9 \times 10^9 \, \text{Nm}^2/\text{C}^2 \) (Coulomb's constant)
- \( R = 0.01 \, \text{m} \) (radius of the ball in meters)

Rearranging gives:
\[ Q = \frac{VR}{k} \]

### Step 6: Substitute the values and solve for \( Q \)
Now we can substitute the values we calculated for \( V \) and \( R \) into the equation for \( Q \).

### Final Calculation
1. Calculate \( E \) in eV.
2. Calculate \( K.E. \).
3. Calculate \( V \).
4. Calculate \( Q \).