An electron moves in a circular orbit with a uniform speed `v`.It produces a magnetic field `B` at the centre of the circle. The radius of the circle is proportional to

Options

A. `(B)/(v)`
B. `(v)/(B)`
C. `sqrt((v)/(B))`
D. `sqrt((B)/(v))`

Answer

To solve the problem of determining the proportionality of the radius of the circular orbit of an electron moving with uniform speed \( v \) and producing a magnetic field \( B \) at the center of the circle, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Current Produced by the Electron:**
- An electron moving in a circular path can be thought of as creating a current. The current \( I \) due to a single electron can be expressed as:
\[ I = \frac{Q}{T} \]
where \( Q \) is the charge of the electron and \( T \) is the time period of one complete revolution.

2. **Charge of the Electron:**
- The charge of an electron is denoted as \( e \). Therefore, we can write:
\[ Q = e \]

3. **Time Period of Circular Motion:**
- The time period \( T \) for the electron moving in a circle of radius \( R \) with speed \( v \) is given by:
\[ T = \frac{2\pi R}{v} \]

4. **Substituting for Current:**
- Now substituting \( Q \) and \( T \) into the equation for current:
\[ I = \frac{e}{\frac{2\pi R}{v}} = \frac{ev}{2\pi R} \]

5. **Magnetic Field at the Center of the Circular Path:**
- The magnetic field \( B \) at the center of the circular path due to the current \( I \) is given by the formula:
\[ B = \frac{\mu_0 I}{2R} \]
- Substituting the expression for \( I \):
\[ B = \frac{\mu_0 \left(\frac{ev}{2\pi R}\right)}{2R} = \frac{\mu_0 ev}{4\pi R} \]

6. **Rearranging for Radius \( R \):**
- Rearranging the equation to express \( R \) in terms of \( B \):
\[ R = \frac{\mu_0 ev}{4\pi B} \]

7. **Identifying Proportionality:**
- From the equation \( R = \frac{\mu_0 ev}{4\pi B} \), we can see that the radius \( R \) is proportional to the ratio of \( v \) to \( B \):
\[ R \propto \frac{v}{B} \]

### Final Answer:
Thus, the radius \( R \) of the circular orbit is proportional to:
\[ R \propto \frac{v}{B} \]