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, we need to analyze the relationship between the radius of the circular orbit of the electron and the magnetic field produced at the center of the circle. Here’s a step-by-step breakdown of the solution:

### Step 1: Define the Current due to the Electron
The electron moving in a circular orbit can be treated as a current loop. The current \( I \) produced by the moving electron can be calculated as:
\[ I = \frac{q}{T} \]
where \( q \) is the charge of the electron (denoted as \( e \)), and \( T \) is the time period of the electron's motion.

### Step 2: Calculate the Time Period \( T \)
The time period \( T \) is the time taken for the electron to complete one full circular orbit. The circumference of the circle is \( 2\pi r \), and since the electron moves with a uniform speed \( v \), we have:
\[ T = \frac{2\pi r}{v} \]

### Step 3: Substitute \( T \) into the Current Formula
Substituting the expression for \( T \) into the current formula gives:
\[ I = \frac{e}{\frac{2\pi r}{v}} = \frac{ev}{2\pi r} \]

### Step 4: Calculate the Magnetic Field \( B \) at the Center
The magnetic field \( B \) at the center of a circular loop carrying current \( I \) is given by the formula:
\[ B = \frac{\mu_0 I}{2r} \]
Substituting the expression for \( I \) from Step 3 into this equation:
\[ B = \frac{\mu_0}{2r} \cdot \frac{ev}{2\pi r} = \frac{\mu_0 ev}{4\pi r^2} \]

### Step 5: Rearranging the Equation to Find \( r \)
From the equation derived in Step 4, we can express \( r \) in terms of \( B \):
\[ B = \frac{\mu_0 ev}{4\pi r^2} \]
Rearranging this gives:
\[ r^2 = \frac{\mu_0 ev}{4\pi B} \]
Taking the square root of both sides:
\[ r = \sqrt{\frac{\mu_0 ev}{4\pi B}} \]

### Step 6: Identify the Proportional Relationship
From the equation \( r = \sqrt{\frac{\mu_0 ev}{4\pi B}} \), we can see that the radius \( r \) is proportional to the square root of the ratio of \( v \) to \( B \):
\[ r \propto \sqrt{\frac{v}{B}} \]

### Conclusion
Thus, the radius \( r \) of the circular orbit of the electron is proportional to the square root of the ratio of the speed \( v \) to the magnetic field \( B \).