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 solve the problem of finding the magnetic field produced at the center of the circle due to an electron revolving in a hydrogen atom, we will follow these steps:

### Step 1: Calculate the Time Period of the Electron
The time period \( T \) of the electron can be calculated using the formula:
\[ T = \frac{2\pi r}{v} \]
where:
- \( r = 5.3 \times 10^{-11} \, \text{m} \) (radius)
- \( v = 7.5 \times 10^4 \, \text{m/s} \) (linear speed)

Substituting the values:
\[ T = \frac{2\pi (5.3 \times 10^{-11})}{7.5 \times 10^4} \]

### Step 2: Calculate the Current in the Loop
The current \( I \) produced by the revolving electron can be calculated using the formula:
\[ I = \frac{e}{T} \]
where:
- \( e = 1.6 \times 10^{-19} \, \text{C} \) (charge of the electron)

Substituting the expression for \( T \):
\[ I = \frac{e \cdot v}{2\pi r} \]

### Step 3: Calculate the Magnetic Field at the Center
The magnetic field \( B \) at the center of the circular path is given by:
\[ B = \frac{\mu_0 I}{2r} \]
where:
- \( \mu_0 = 4\pi \times 10^{-7} \, \text{T m/A} \) (permeability of free space)

Substituting the expression for \( I \):
\[ B = \frac{\mu_0 \cdot \left(\frac{e \cdot v}{2\pi r}\right)}{2r} \]

### Step 4: Substitute All Values and Calculate B
Now we can substitute all known values into the equation:
\[ B = \frac{4\pi \times 10^{-7} \cdot \left(\frac{1.6 \times 10^{-19} \cdot 7.5 \times 10^4}{2\pi (5.3 \times 10^{-11})}\right)}{2(5.3 \times 10^{-11})} \]

### Step 5: Simplify and Calculate
After substituting the values, we can simplify the expression to find the magnetic field \( B \).

### Final Calculation
Calculating the above expression will yield:
\[ B \approx 0.43 \, \text{T} \, \text{(or Weber/m}^2\text{)} \]

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

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