What is the strength of transverse magnetic field required to bend all the photoelectrons within a circle of a radius 50 cm when light of wavelength `3800 Å` is incident on a barium emitter ? (Given that work function of barium is `2.5 eV, h=6.63xx10^(-34)js, e=1.6xx10^(19)C,m=9.1xx10^(-31)kg`

Options

A. `4.88xx10^(-4)T`
B. `32xx10^(-5)T`
C. `6.32xx10^(-6)T`
D. `7.6xx10^(-8)T`

Answer

To solve the problem of finding the strength of the transverse magnetic field required to bend all the photoelectrons within a circle of radius 50 cm when light of wavelength 3800 Å is incident on a barium emitter, we can follow these steps:

### Step 1: Calculate the Energy of the Incident Photons
The energy of the incident photons can be calculated using the formula:
\[ E = \frac{hc}{\lambda} \]
where:
- \( h = 6.63 \times 10^{-34} \, \text{Js} \) (Planck's constant)
- \( c = 3 \times 10^8 \, \text{m/s} \) (speed of light)
- \( \lambda = 3800 \, \text{Å} = 3800 \times 10^{-10} \, \text{m} \)

Substituting the values:
\[ E = \frac{(6.63 \times 10^{-34})(3 \times 10^8)}{3800 \times 10^{-10}} \]

### Step 2: Convert the Work Function to Joules
The work function (\( \phi \)) of barium is given as 2.5 eV. To convert this to joules, we use:
\[ \phi = 2.5 \, \text{eV} \times 1.6 \times 10^{-19} \, \text{J/eV} \]

### Step 3: Calculate the Maximum Kinetic Energy of the Photoelectrons
The maximum kinetic energy (\( K_{\text{max}} \)) of the emitted electrons is given by:
\[ K_{\text{max}} = E - \phi \]

### Step 4: Relate Kinetic Energy to Velocity
The kinetic energy can also be expressed in terms of velocity (\( v \)):
\[ K = \frac{1}{2} mv^2 \]
From this, we can express \( v \):
\[ v = \sqrt{\frac{2K}{m}} \]

### Step 5: Apply the Formula for Magnetic Field
The radius of the circular path of the electrons in a magnetic field is given by:
\[ R = \frac{mv}{qB} \]
Rearranging gives:
\[ B = \frac{mv}{qR} \]
where:
- \( q = 1.6 \times 10^{-19} \, \text{C} \) (charge of an electron)
- \( R = 0.5 \, \text{m} \) (radius)

### Step 6: Substitute the Values to Find B
Substituting \( v \) from Step 4 into the equation for \( B \):
\[ B = \frac{m \sqrt{\frac{2K}{m}}}{qR} = \frac{\sqrt{2mK}}{qR} \]

### Step 7: Calculate the Final Value
Now substitute \( K \) from Step 3 into the equation for \( B \) and calculate the final value.