An electron beam accelerated from rest through a potential difference of 5000 V in vacuum is allowed to impinge on a surface normally. The incident current is 50`muA` and if the electrons come to rest on striking the surface the force on it is

Options

A. `1.1924xx10^-8N`
B. `2.1xx10^-8N`
C. `1.6xx10^-8N`
D. `1.6xx10^-6N`

Explanation

To solve the problem step-by-step, we will follow the logical sequence of calculations as outlined in the video transcript.

### Step 1: Calculate the Energy of the Electrons
The energy gained by an electron when accelerated through a potential difference (V) is given by:
\[ E = eV \]
where:
- \( e \) is the charge of an electron (\( 1.6 \times 10^{-19} \) coulombs),
- \( V \) is the potential difference (5000 V).

Substituting the values:
\[ E = 1.6 \times 10^{-19} \, \text{C} \times 5000 \, \text{V} = 8.0 \times 10^{-16} \, \text{J} \]

### Step 2: Relate Energy to Momentum
The kinetic energy of the electrons can also be expressed in terms of momentum (p):
\[ E = \frac{p^2}{2m} \]
where:
- \( p = mv \) (momentum),
- \( m \) is the mass of an electron (\( 9.1 \times 10^{-31} \) kg).

Rearranging gives:
\[ p^2 = 2mE \]
Thus,
\[ p = \sqrt{2mE} \]

### Step 3: Calculate the Momentum
Substituting the values of \( E \) and \( m \):
\[ p = \sqrt{2 \times (9.1 \times 10^{-31} \, \text{kg}) \times (8.0 \times 10^{-16} \, \text{J})} \]
Calculating this gives:
\[ p \approx \sqrt{1.456 \times 10^{-46}} \approx 1.208 \times 10^{-23} \, \text{kg m/s} \]

### Step 4: Calculate the Number of Electrons Striking the Surface
The number of electrons striking the surface per second (n) can be calculated using the current (I):
\[ n = \frac{I}{e} \]
where:
- \( I = 50 \, \mu A = 50 \times 10^{-6} \, A \).

Substituting the values:
\[ n = \frac{50 \times 10^{-6}}{1.6 \times 10^{-19}} \approx 3.125 \times 10^{14} \, \text{electrons/s} \]

### Step 5: Calculate the Force on the Surface
The force (F) imparted by the electrons striking the surface can be calculated using the change in momentum:
\[ F = n \cdot \Delta p \]
Since the electrons come to rest upon striking the surface, the change in momentum is equal to the momentum of the electrons:
\[ F = n \cdot p \]

Substituting the values:
\[ F = (3.125 \times 10^{14}) \cdot (1.208 \times 10^{-23}) \]
Calculating this gives:
\[ F \approx 3.78 \times 10^{-9} \, \text{N} \]

### Final Answer
Thus, the force on the surface due to the electron beam is approximately:
\[ F \approx 3.78 \times 10^{-9} \, \text{N} \]

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