What is the magnetic energy density (in terms of standard constant & `r` )at the centre of a circulating electron in the hydrogen atom in first orbit.(Radius of the orbit is `r`)

Explanation

To find the magnetic energy density at the center of a circulating electron in a hydrogen atom in its first orbit, we can follow these steps:

### Step 1: Understand the System
The electron in a hydrogen atom revolves around the nucleus (proton) in a circular orbit. The centripetal force required for this circular motion is provided by the electrostatic force of attraction between the negatively charged electron and the positively charged nucleus.

### Step 2: Determine the Current Due to the Electron
The current \( I \) due to the circulating electron can be expressed as:
\[ I = \frac{e}{T} \]
where \( e \) is the charge of the electron and \( T \) is the time period of one complete revolution.

### Step 3: Calculate the Time Period \( T \)
The time period \( T \) can be calculated as:
\[ T = \frac{\text{Distance}}{\text{Orbital Velocity}} = \frac{2\pi r}{V} \]
where \( r \) is the radius of the orbit and \( V \) is the orbital velocity of the electron.

### Step 4: Substitute \( T \) into the Current Formula
Substituting \( T \) into the current formula gives:
\[ I = \frac{eV}{2\pi r} \]

### Step 5: Calculate the Magnetic Field \( B \) at the Center
The magnetic field \( B \) at the center of the loop (the orbit of the electron) can be given by:
\[ B = \frac{\mu_0 I}{2r} \]
where \( \mu_0 \) is the permeability of free space.

### Step 6: Substitute \( I \) into the Magnetic Field Formula
Substituting the expression for \( I \) into the formula for \( B \):
\[ B = \frac{\mu_0}{2r} \cdot \frac{eV}{2\pi r} = \frac{\mu_0 eV}{4\pi r^2} \]

### Step 7: Calculate the Magnetic Energy Density
The magnetic energy density \( u \) is given by:
\[ u = \frac{B^2}{2\mu_0} \]
Substituting the expression for \( B \):
\[ u = \frac{1}{2\mu_0} \left(\frac{\mu_0 eV}{4\pi r^2}\right)^2 \]

### Step 8: Simplify the Expression
Simplifying the expression:
\[ u = \frac{1}{2\mu_0} \cdot \frac{\mu_0^2 e^2 V^2}{16\pi^2 r^4} = \frac{\mu_0 e^2 V^2}{32\pi^2 r^4} \]

### Final Answer
Thus, the magnetic energy density at the center of a circulating electron in a hydrogen atom in the first orbit is:
\[ u = \frac{\mu_0 e^2 V^2}{32\pi^2 r^4} \]

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