Find an expression fot the magneitc dipole moment and magnetic field induction at the centre of Bohr'r hypothetical hydrogen atom in the n th orbit of the electron in terms of universal constant.

Explanation

To find the expressions for the magnetic dipole moment and the magnetic field induction at the center of Bohr's hypothetical hydrogen atom in the nth orbit of the electron, we can follow these steps:

### Step 1: Magnetic Dipole Moment Calculation

1. **Formula for Magnetic Dipole Moment (μ)**:
The magnetic dipole moment for an electron in the nth orbit can be expressed as:
\[ \mu_n = I \cdot A \]
where \(I\) is the current and \(A\) is the area of the orbit.

2. **Current (I) in nth Orbit**:
The current \(I\) can be defined as the charge \(Q\) passing through the orbit per unit time. The time period \(T_n\) for one complete revolution of the electron in the nth orbit is given by:
\[ I = \frac{Q}{T_n} \]

3. **Area (A) of nth Orbit**:
The area \(A\) of the circular orbit is:
\[ A = \pi R_n^2 \]
where \(R_n\) is the radius of the nth orbit.

4. **Substituting for Current and Area**:
Therefore, the magnetic dipole moment can be rewritten as:
\[ \mu_n = \left(\frac{Q}{T_n}\right) \cdot (\pi R_n^2) \]

5. **Finding the Time Period (T_n)**:
The time period \(T_n\) can be expressed in terms of angular velocity (\(\omega_n\)):
\[ T_n = \frac{2\pi}{\omega_n} \]
Thus, we can substitute this into the current expression:
\[ I = \frac{Q \cdot \omega_n}{2\pi} \]

6. **Final Expression for Magnetic Dipole Moment**:
Substituting \(I\) back into the expression for \(\mu_n\):
\[ \mu_n = \frac{Q \cdot \omega_n}{2\pi} \cdot \pi R_n^2 = \frac{Q \cdot \omega_n \cdot R_n^2}{2} \]

7. **Expressing in Terms of Constants**:
Using the relationships \(V_n = \frac{Z e^2}{n}\) and \(R_n = \frac{R_0 n^2}{Z}\), we can simplify further:
\[ \mu_n = \frac{Q \cdot V_n \cdot R_n}{2} \]
This leads to:
\[ \mu_n = \frac{Q \cdot V_0 \cdot R_0}{2} \cdot n \]
where \(V_0\) and \(R_0\) are constants.

### Step 2: Magnetic Field Induction Calculation

1. **Formula for Magnetic Field (B)**:
The magnetic field induction at the center of the atom due to the revolving electron is given by:
\[ B_n = \frac{\mu_0 I_n}{2 R_n} \]

2. **Substituting for Current (I_n)**:
Using the expression for current \(I_n\):
\[ B_n = \frac{\mu_0 \cdot \frac{Q}{T_n}}{2 R_n} \]

3. **Substituting for Time Period (T_n)**:
Substitute \(T_n\) in the expression:
\[ B_n = \frac{\mu_0 \cdot Q \cdot \omega_n}{4 \pi R_n} \]

4. **Final Expression for Magnetic Field**:
Substituting \(R_n\) and simplifying:
\[ B_n = \frac{\mu_0 Q V_0 Z^3}{4 \pi R_0^2 n^5} \]
This can be expressed as:
\[ B_n = B_0 \frac{Z^3}{n^5} \]
where \(B_0\) is a constant.

### Summary of Results
- **Magnetic Dipole Moment**:
\[ \mu_n = \frac{Q V_0 R_0}{2} n \]
- **Magnetic Field Induction**:
\[ B_n = B_0 \frac{Z^3}{n^5} \]