Photons of wavelength `6556 A^@` falls on a metal surface. If ejected electrons with maximum K.E. moves in magnetic field of `3 xx 10^(-4) T` in circular orbit of radius `10^(-2)` m, then work function of metal surface is

Options

A. 1.8 eV
B. 0.8 eV
C. 1.1 eV
D. 1.4 eV

Explanation

To solve the problem, we will follow these steps:

### Step 1: Calculate the energy of the incoming photons
The energy of a 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)
- \( \lambda = 6556 \, \text{Å} = 6556 \times 10^{-10} \, \text{m} \)

Substituting the values:
\[ E = \frac{(6.626 \times 10^{-34})(3 \times 10^8)}{6556 \times 10^{-10}} \]

### Step 2: Calculate the maximum kinetic energy of the ejected electrons
The maximum kinetic energy (K.E.) of the ejected electrons can be expressed as:
\[ K.E. = \frac{1}{2} mv^2 \]
Where \( m \) is the mass of the electron and \( v \) is its velocity. The velocity can be derived from the radius of the circular motion in the magnetic field:
\[ r = \frac{mv}{qB} \]
Rearranging gives:
\[ mv = qBr \]
Substituting this into the kinetic energy formula:
\[ K.E. = \frac{1}{2} \frac{(qBr)^2}{m} \]

### Step 3: Substitute known values
Where:
- \( q = 1.6 \times 10^{-19} \, \text{C} \) (charge of the electron)
- \( B = 3 \times 10^{-4} \, \text{T} \) (magnetic field)
- \( r = 10^{-2} \, \text{m} \) (radius)
- \( m = 9.11 \times 10^{-31} \, \text{kg} \) (mass of the electron)

Substituting these values into the kinetic energy formula:
\[ K.E. = \frac{1}{2} \frac{(1.6 \times 10^{-19})(3 \times 10^{-4})(10^{-2})^2}{9.11 \times 10^{-31}} \]

### Step 4: Calculate the work function
The work function \( \phi \) can be calculated using the equation:
\[ \phi = E - K.E. \]
Where \( E \) is the energy of the incoming photon calculated in Step 1 and \( K.E. \) is calculated in Step 3.

### Step 5: Final calculations
After calculating \( E \) and \( K.E. \), substitute these values into the work function equation to find \( \phi \).