Calculate the value of `X` if magnetic field strength at the centre of a hydrogen atom caused by an electron moving along the first Bohr orbits is `(X)/(2)T`.

Explanation

To calculate the value of `X` in the magnetic field strength at the center of a hydrogen atom caused by an electron moving along the first Bohr orbit, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Current Due to Electron Motion**:
The electron moving in a circular path can be treated as a current loop. The current `I` due to the electron can be expressed as:
\[ I = \frac{Q}{T} \]
where \( Q \) is the charge of the electron and \( T \) is the time period for one complete revolution.

2. **Calculating the Time Period**:
The time period \( T \) for the electron to complete one revolution is given by:
\[ T = \frac{2\pi r}{v} \]
where \( r \) is the radius of the orbit and \( v \) is the velocity of the electron.

3. **Substituting the Time Period into Current Formula**:
Substituting \( T \) into the current formula, we have:
\[ I = \frac{e}{T} = \frac{e}{\frac{2\pi r}{v}} = \frac{ev}{2\pi r} \]
where \( e \) is the charge of the electron (approximately \( 1.6 \times 10^{-19} \, C \)).

4. **Magnetic Field at the Center of the Loop**:
The magnetic field \( B \) at the center of a circular loop carrying current \( I \) is given by:
\[ B = \frac{\mu_0 I}{2r} \]
where \( \mu_0 \) is the permeability of free space (\( \mu_0 = 4\pi \times 10^{-7} \, T \cdot m/A \)).

5. **Substituting Current into Magnetic Field Formula**:
Substituting the expression for \( I \) into the magnetic field formula, we get:
\[ B = \frac{\mu_0 \left(\frac{ev}{2\pi r}\right)}{2r} = \frac{\mu_0 ev}{4\pi r^2} \]

6. **Using Known Values**:
For the first Bohr orbit:
- Charge of electron \( e = 1.6 \times 10^{-19} \, C \)
- Velocity of electron \( v = 2.2 \times 10^6 \, m/s \)
- Radius of the first Bohr orbit \( r = 0.52 \times 10^{-10} \, m \)

Now substituting these values into the magnetic field equation:
\[ B = \frac{(4\pi \times 10^{-7}) \cdot (1.6 \times 10^{-19}) \cdot (2.2 \times 10^6)}{4\pi (0.52 \times 10^{-10})^2} \]

7. **Calculating the Magnetic Field**:
Simplifying the equation:
\[ B = \frac{(10^{-7}) \cdot (1.6 \times 10^{-19}) \cdot (2.2 \times 10^6)}{(0.52 \times 10^{-10})^2} \]
After calculating, you will find:
\[ B \approx 13 \, T \]

8. **Expressing in Terms of X**:
The problem states that the magnetic field can be expressed as:
\[ B = \frac{X}{2} \, T \]
Setting \( 13 = \frac{X}{2} \) leads to:
\[ X = 26 \]

### Final Answer:
The value of \( X \) is \( 26 \).