Find an expression for the magnetic dipole moment and magnetic field induction at the center of Bohr's hypothetical hydrogen atom in the nth orbit of the electron in terms of universal constant.

Explanation

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

### Step 1: Magnetic Dipole Moment
The magnetic dipole moment (M) of the electron in the nth orbit can be expressed in terms of its angular momentum (L). According to Bohr's model, the angular momentum of the electron is given by:

\[ L = \frac{n h}{2\pi} \]

where:
- \( n \) is the principal quantum number,
- \( h \) is Planck's constant.

The magnetic dipole moment is related to angular momentum by the formula:

\[ M = \frac{L Q}{2m} \]

where:
- \( Q \) is the charge of the electron (denoted as \( e \)),
- \( m \) is the mass of the electron.

Substituting the expression for \( L \):

\[ M = \frac{\left(\frac{n h}{2\pi}\right) e}{2m} \]

This simplifies to:

\[ M = \frac{n h e}{4\pi m} \]

### Step 2: Magnetic Field Induction
To find the magnetic field induction (B) at the center of the hydrogen atom, we can use the formula:

\[ B = \frac{\mu_0 I}{2R} \]

where:
- \( \mu_0 \) is the permeability of free space,
- \( I \) is the current due to the orbiting electron,
- \( R \) is the radius of the nth orbit.

The current \( I \) can be expressed as:

\[ I = Qf \]

where \( f \) is the frequency of revolution of the electron. The frequency can be related to the velocity \( V \) and radius \( R \) of the orbit:

\[ f = \frac{V}{2\pi R} \]

Thus, we can write:

\[ I = e \cdot \frac{V}{2\pi R} \]

Now, we need to find expressions for \( V \) and \( R \). From Bohr's model, we know:

\[ \frac{mV^2}{R} = \frac{1}{4\pi \epsilon_0} \cdot \frac{e^2}{R^2} \]

This can be rearranged to find \( V \):

\[ V = \frac{e^2}{2\epsilon_0 n h} \]

And for the radius \( R \):

\[ R = \frac{\epsilon_0 n^2 h^2}{\pi m e^2} \]

### Step 3: Substitute into B
Substituting \( I \) into the equation for \( B \):

\[ B = \frac{\mu_0}{2R} \cdot e \cdot \frac{V}{2\pi R} \]

Substituting the expressions for \( V \) and \( R \):

\[ B = \frac{\mu_0 e}{4\pi R^2} \cdot \frac{e^2}{2\epsilon_0 n h} \]

After substituting \( R \) and simplifying, we arrive at:

\[ B = \frac{\mu_0 \pi m e^7}{8 \epsilon_0^3 h^5 n^5} \]

### Final Expressions
Thus, the final expressions are:
1. Magnetic dipole moment:
\[ M = \frac{n h e}{4\pi m} \]
2. Magnetic field induction:
\[ B = \frac{\mu_0 \pi m e^7}{8 \epsilon_0^3 h^5 n^5} \]