An electron of mass m,moves around the nucleus in a circular orbit of radius 'r' under the action of centripetal force 'F'. The equivalent electric current is

Options

A. `(e )/(2 pi ) sqrt((F )/(mr ))`
B. ` 2pi Sqrt(( F )/(mr))`
C. `(e )/( pi ) sqrt((F )/(mr))`
D. `(e )/( 2 pi ) sqrt((mr )/( F )`

Answer

To find the equivalent electric current for an electron moving in a circular orbit around a nucleus under the action of centripetal force, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of Centripetal Force**:
The centripetal force \( F \) acting on the electron can be expressed as:
\[ F = \frac{mv^2}{r} \]
where \( m \) is the mass of the electron, \( v \) is its velocity, and \( r \) is the radius of the circular orbit.

2. **Determine the Charge of the Electron**:
The charge \( q \) of the electron is denoted as \( e \).

3. **Define the Current**:
The electric current \( I \) is defined as the charge per unit time:
\[ I = \frac{q}{t} \]
Substituting the charge of the electron, we have:
\[ I = \frac{e}{t} \]

4. **Calculate the Time Period \( t \)**:
The time period \( t \) for one complete revolution of the electron can be calculated using the circumference of the circular path divided by the velocity:
\[ t = \frac{\text{Circumference}}{\text{Velocity}} = \frac{2\pi r}{v} \]

5. **Express Velocity in Terms of Centripetal Force**:
From the centripetal force equation, we can express the velocity \( v \):
\[ v = \sqrt{\frac{Fr}{m}} \]

6. **Substitute Velocity into the Time Period Equation**:
Substituting the expression for \( v \) into the equation for \( t \):
\[ t = \frac{2\pi r}{\sqrt{\frac{Fr}{m}}} \]
Simplifying this gives:
\[ t = 2\pi \sqrt{\frac{m r}{F}} \]

7. **Substitute Time Period Back into the Current Equation**:
Now substituting \( t \) back into the equation for current \( I \):
\[ I = \frac{e}{2\pi \sqrt{\frac{m r}{F}}} \]
This can be rearranged to:
\[ I = \frac{e}{2\pi} \sqrt{\frac{F}{m r}} \]

8. **Final Expression for Equivalent Electric Current**:
Thus, the equivalent electric current \( I \) is given by:
\[ I = \frac{e}{2\pi} \sqrt{\frac{F}{m r}} \]

### Conclusion:
The equivalent electric current is:
\[ I = \frac{e}{2\pi} \sqrt{\frac{F}{m r}} \]