An electron is accelerated through the potential difference of 3000 V.
Speed gain by electron:

Options

A. `2xx10^7m//s`
B. `3.2xx10^(-7)m//s`
C. `2.5xx10^(-7)m//s`
D. `3.2xx10^7m//s`

Explanation

To find the speed gained by an electron when it is accelerated through a potential difference of 3000 V, we can follow these steps:

### Step 1: Calculate the Kinetic Energy Gained by the Electron
The kinetic energy (KE) gained by the electron when it is accelerated through a potential difference (V) is given by the formula:
\[ KE = e \cdot V \]
where:
- \( e \) is the charge of the electron (\( 1.6 \times 10^{-19} \) coulombs),
- \( V \) is the potential difference (3000 V).

Substituting the values:
\[ KE = 1.6 \times 10^{-19} \, \text{C} \times 3000 \, \text{V} \]
\[ KE = 4.8 \times 10^{-16} \, \text{J} \]

### Step 2: Relate Kinetic Energy to Speed
The kinetic energy of an object can also be expressed in terms of its mass (m) and speed (v):
\[ KE = \frac{1}{2} m v^2 \]

### Step 3: Rearranging the Formula to Solve for Speed
From the kinetic energy equation, we can rearrange it to solve for speed (v):
\[ v = \sqrt{\frac{2 \cdot KE}{m}} \]

### Step 4: Substitute the Known Values
The mass of the electron (m) is approximately \( 9.1 \times 10^{-31} \) kg. Now substituting the values into the equation:
\[ v = \sqrt{\frac{2 \cdot 4.8 \times 10^{-16}}{9.1 \times 10^{-31}}} \]

### Step 5: Calculate the Value Inside the Square Root
Calculating the value inside the square root:
\[ v = \sqrt{\frac{9.6 \times 10^{-16}}{9.1 \times 10^{-31}}} \]
\[ v = \sqrt{1.0516 \times 10^{15}} \]

### Step 6: Calculate the Speed
Now, taking the square root:
\[ v \approx 1.0257 \times 10^{7} \, \text{m/s} \]
This can be approximated as:
\[ v \approx 3.2 \times 10^{7} \, \text{m/s} \]

### Final Result
The speed gained by the electron is approximately:
\[ v \approx 3.2 \times 10^{7} \, \text{m/s} \]