An electron is moving with a velocity of `10^(7) m s^(-1)`. Find its energy in electron volts. (Mass of electron `=9.1xx10^(-31) kg` and charge of electron `=1.6xx10^(-19) C`)

Answer

To find the energy of an electron moving with a velocity of \(10^7 \, \text{m/s}\), we will calculate its kinetic energy using the formula:

\[ KE = \frac{1}{2} mv^2 \]

where:
- \(m\) is the mass of the electron,
- \(v\) is the velocity of the electron.

### Step 1: Substitute the values into the kinetic energy formula

Given:
- Mass of the electron, \(m = 9.1 \times 10^{-31} \, \text{kg}\)
- Velocity of the electron, \(v = 10^7 \, \text{m/s}\)

Substituting these values into the kinetic energy formula:

\[ KE = \frac{1}{2} \times (9.1 \times 10^{-31} \, \text{kg}) \times (10^7 \, \text{m/s})^2 \]

### Step 2: Calculate \(v^2\)

Calculating \(v^2\):

\[ (10^7 \, \text{m/s})^2 = 10^{14} \, \text{m}^2/\text{s}^2 \]

### Step 3: Substitute \(v^2\) back into the kinetic energy formula

Now substituting \(v^2\) back into the equation:

\[ KE = \frac{1}{2} \times (9.1 \times 10^{-31} \, \text{kg}) \times (10^{14} \, \text{m}^2/\text{s}^2) \]

### Step 4: Calculate the kinetic energy in joules

Calculating the kinetic energy:

\[ KE = \frac{1}{2} \times 9.1 \times 10^{-31} \times 10^{14} \]

Calculating \( \frac{1}{2} \times 9.1 = 4.55\):

\[ KE = 4.55 \times 10^{-31 + 14} = 4.55 \times 10^{-17} \, \text{J} \]

### Step 5: Convert joules to electron volts

To convert joules to electron volts, we use the conversion factor:

\[ 1 \, \text{eV} = 1.6 \times 10^{-19} \, \text{J} \]

Now, we can convert the energy from joules to electron volts:

\[ KE \text{ (in eV)} = \frac{4.55 \times 10^{-17} \, \text{J}}{1.6 \times 10^{-19} \, \text{J/eV}} \]

### Step 6: Perform the division

Calculating the division:

\[ KE \text{ (in eV)} = \frac{4.55}{1.6} \times 10^{-17 + 19} = \frac{4.55}{1.6} \times 10^{2} \]

Calculating \( \frac{4.55}{1.6} \approx 2.84\):

\[ KE \text{ (in eV)} \approx 2.84 \times 10^{2} = 284 \, \text{eV} \]

### Final Answer

The energy of the electron moving with a velocity of \(10^7 \, \text{m/s}\) is approximately **284 eV**.

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