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

Answer

To find the equivalent electric current \( I \) for an electron moving in a circular orbit around a nucleus, we can follow these steps:

### Step 1: Understand the Concept of Electric Current
Electric current \( I \) is defined as the flow of charge per unit time. Mathematically, it can be expressed as:
\[ I = \frac{Q}{t} \]
where \( Q \) is the charge and \( t \) is the time.

### Step 2: Determine the Charge and Time Period
For an electron, the charge \( Q \) is equal to the elementary charge \( e \). The time period \( T \) for one complete revolution in a circular orbit of radius \( r \) can be expressed in terms of the velocity \( v \):
\[ T = \frac{2\pi r}{v} \]
Thus, the current can be rewritten as:
\[ I = \frac{Q}{T} = \frac{e}{T} = \frac{e \cdot v}{2\pi r} \]

### Step 3: Relate Velocity to Centripetal Force
The centripetal force \( F \) acting on the electron is given by:
\[ F = \frac{mv^2}{r} \]
From this, we can express the velocity \( v \) in terms of \( F \):
\[ v^2 = \frac{Fr}{m} \quad \Rightarrow \quad v = \sqrt{\frac{Fr}{m}} \]

### Step 4: Substitute Velocity into the Current Equation
Substituting the expression for \( v \) into the current equation:
\[ I = \frac{e \cdot \sqrt{\frac{Fr}{m}}}{2\pi r} \]

### Step 5: Simplify the Expression
Now, simplifying the expression for \( I \):
\[ I = \frac{e}{2\pi r} \cdot \sqrt{\frac{Fr}{m}} = \frac{e}{2\pi} \cdot \sqrt{\frac{F}{m}} \cdot \frac{1}{\sqrt{r}} \]

### Final Expression
Thus, the equivalent electric current \( I \) is given by:
\[ I = \frac{e}{2\pi} \sqrt{\frac{F}{m}} \cdot \frac{1}{\sqrt{r}} \]