Calcualte 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 \( X \) for the magnetic field strength at the center of a hydrogen atom caused by an electron moving along the first Bohr orbit, we will follow these steps:

### Step 1: Understand the parameters
The magnetic field strength \( B \) at the center of a hydrogen atom due to an electron in circular motion can be expressed using the formula:
\[ B = \frac{\mu_0 I}{2r} \]
where:
- \( \mu_0 \) is the permeability of free space,
- \( I \) is the effective current due to the electron,
- \( r \) is the radius of the orbit.

### Step 2: Calculate the effective current \( I \)
The effective current \( I \) can be calculated from the charge of the electron and its orbital frequency. The current due to a moving charge is given by:
\[ I = \frac{q}{T} \]
where \( q \) is the charge of the electron and \( T \) is the time period of its motion.

For an electron in the first Bohr orbit:
- The charge \( q = e = 1.6 \times 10^{-19} \, C \).
- The time period \( T \) can be related to the frequency \( f \) as \( T = \frac{1}{f} \).

### Step 3: Determine the radius \( r \)
The radius of the first Bohr orbit is given by:
\[ r = 0.529 \times 10^{-10} \, m \]

### Step 4: Substitute values into the magnetic field equation
Substituting the values into the magnetic field equation:
\[ B = \frac{\mu_0 I}{2r} \]
We know that:
\[ I = \frac{e}{T} \]
Thus:
\[ B = \frac{\mu_0 e}{2rT} \]

### Step 5: Substitute known values
Using \( \mu_0 = 4\pi \times 10^{-7} \, T \cdot m/A \) and substituting \( r \):
\[ B = \frac{4\pi \times 10^{-7} \cdot 1.6 \times 10^{-19}}{2 \cdot 0.529 \times 10^{-10} \cdot T} \]

### Step 6: Simplify the equation
Now, we simplify the equation to find \( B \):
\[ B = \frac{4\pi \times 1.6}{2 \cdot 0.529} \times \frac{10^{-7} \times 10^{-19}}{10^{-10} \cdot T} \]
\[ B = \frac{3.2\pi \times 10^{-26}}{1.058 \cdot T} \]

### Step 7: Set up the equation for \( B \)
We are given that:
\[ B = \frac{x}{2} T \]
Equating both expressions for \( B \):
\[ \frac{3.2\pi \times 10^{-26}}{1.058 \cdot T} = \frac{x}{2} T \]

### Step 8: Solve for \( x \)
Cross-multiplying gives:
\[ 3.2\pi \times 10^{-26} = \frac{x}{2} \cdot 1.058 T^2 \]
Rearranging for \( x \):
\[ x = \frac{3.2\pi \times 10^{-26} \cdot 2}{1.058 T^2} \]

### Step 9: Final calculation
Assuming \( T \) is known or can be calculated, we can find \( x \).

### Conclusion
The final value of \( x \) will depend on the specific value of \( T \) used in the calculations.

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