An electron entering field normally with a velocity `4 xx 10^7 ms^(-1)` travels a distance of 0.10 m in an electric field of intensity `3200 Vm^(-1)`. What is the deviation from its path?

Options

A. `1.76 mm`
B. `17.6 mm`
C. `176 mm`
D. `0.176 mm`

Answer

To solve the problem of finding the deviation of an electron entering an electric field, we can follow these steps:

### Step 1: Determine the time taken by the electron to travel through the electric field.

The electron travels a distance \( S_x = 0.1 \, m \) with a velocity \( V_x = 4 \times 10^7 \, m/s \).

Using the formula for time:
\[ t = \frac{S_x}{V_x} \]
Substituting the values:
\[ t = \frac{0.1}{4 \times 10^7} = 2.5 \times 10^{-9} \, s \]

### Step 2: Calculate the electric force acting on the electron.

The electric force \( F \) on the electron can be calculated using:
\[ F = qE \]
where \( q \) is the charge of the electron (\( q = 1.6 \times 10^{-19} \, C \)) and \( E \) is the electric field intensity (\( E = 3200 \, V/m \)).

Substituting the values:
\[ F = (1.6 \times 10^{-19})(3200) = 5.12 \times 10^{-16} \, N \]

### Step 3: Calculate the acceleration of the electron.

Using Newton's second law, the acceleration \( a \) can be calculated as:
\[ a = \frac{F}{m} \]
where \( m \) is the mass of the electron (\( m = 9.1 \times 10^{-31} \, kg \)).

Substituting the values:
\[ a = \frac{5.12 \times 10^{-16}}{9.1 \times 10^{-31}} \approx 5.62 \times 10^{14} \, m/s^2 \]

### Step 4: Calculate the deviation in the y-direction.

The deviation \( S_y \) (which is the distance moved in the y-direction) can be calculated using the formula:
\[ S_y = \frac{1}{2} a t^2 \]

Substituting the values:
\[ S_y = \frac{1}{2} (5.62 \times 10^{14}) (2.5 \times 10^{-9})^2 \]
Calculating \( t^2 \):
\[ t^2 = (2.5 \times 10^{-9})^2 = 6.25 \times 10^{-18} \]
Now substituting:
\[ S_y = \frac{1}{2} (5.62 \times 10^{14}) (6.25 \times 10^{-18}) \approx 1.76 \times 10^{-3} \, m \]

### Step 5: Convert the deviation to millimeters.

To convert meters to millimeters:
\[ S_y = 1.76 \times 10^{-3} \, m = 1.76 \, mm \]

### Final Answer:
The deviation from its path is \( 1.76 \, mm \).

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