Imagine that an electron revolves around a circle of the radius `5.3xx10^(-11)m` with a linear speed of `7.5xx10^4 ms^(-1)` in a hydrogen atom. The magnetic field produced at the centre of the circle, due to the electron, is

Options

A. `43Wbm^(-2)`
B. `4300Wbm^(-2)`
C. `0.43Wbm^(-2)`
D. `43xx10^(-4)Wbm^(-2)`

Answer

To find the magnetic field produced at the center of the circle due to an electron revolving around it, we can follow these steps:

### Step 1: Understand the parameters
We are given:
- Radius of the circle, \( r = 5.3 \times 10^{-11} \, \text{m} \)
- Linear speed of the electron, \( v = 7.5 \times 10^4 \, \text{m/s} \)
- Charge of the electron, \( Q = 1.6 \times 10^{-19} \, \text{C} \)

### Step 2: Calculate the time period (T)
The time period \( T \) is the time taken for one complete revolution. It can be calculated using the formula:
\[ T = \frac{2\pi r}{v} \]
Substituting the values:
\[ T = \frac{2\pi (5.3 \times 10^{-11})}{7.5 \times 10^4} \]

### Step 3: Calculate the current (I)
The current \( I \) due to the moving charge is given by:
\[ I = \frac{Q}{T} \]
Substituting \( T \) from the previous calculation:
\[ I = \frac{Q \cdot v}{2\pi r} \]

### Step 4: Calculate the magnetic field (B)
The magnetic field \( B \) at the center of the circular path due to the current can be calculated using the formula:
\[ B = \frac{\mu_0 I}{2r} \]
Where \( \mu_0 = 4\pi \times 10^{-7} \, \text{T m/A} \).

### Step 5: Substitute the values into the magnetic field formula
Substituting the expression for \( I \) into the magnetic field formula:
\[ B = \frac{\mu_0 \cdot \frac{Q \cdot v}{2\pi r}}{2r} \]
This simplifies to:
\[ B = \frac{\mu_0 Q v}{4\pi r^2} \]

### Step 6: Plug in the values
Now substituting the known values:
\[ B = \frac{(4\pi \times 10^{-7}) \cdot (1.6 \times 10^{-19}) \cdot (7.5 \times 10^4)}{4\pi (5.3 \times 10^{-11})^2} \]
The \( 4\pi \) cancels out:
\[ B = \frac{(1.6 \times 10^{-19}) \cdot (7.5 \times 10^4)}{(5.3 \times 10^{-11})^2} \]

### Step 7: Calculate the final value
Calculating the above expression gives:
\[ B \approx 0.43 \, \text{T} \]

Thus, the magnetic field produced at the center of the circle due to the electron is approximately \( 0.43 \, \text{T} \).