The momentum of an electron in an orbit is `h//lambda`, where h is a constant and `lambda` is wavelength associated with it. The nuclear magneton of electron of charge e and mass`m_(e)` is given as `mu_(n)=(eh)/(3672 pim_(e))`. The dimension of `mu_(n)` are `(Ararr"current")`

Options

A. `[ML^(2)A]`
B. `[ML^(3)A]`
C. `[L^(2)A]`
D. `[ML^(2)]`

Explanation

To find the dimensions of the nuclear magneton \(\mu_n\) of an electron, we will follow these steps:

### Step 1: Understand the given formula
The nuclear magneton \(\mu_n\) is given by the formula:
\[ \mu_n = \frac{eh}{3672 \pi m_e} \]
where:
- \(e\) is the charge of the electron,
- \(h\) is Planck's constant,
- \(m_e\) is the mass of the electron.

### Step 2: Determine the dimensions of each component
1. **Charge (\(e\))**: The dimension of electric charge is given as:
\[ [e] = [I][T] = A \cdot s \]
where \(A\) is the unit of current (Ampere) and \(s\) is the unit of time.

2. **Planck's constant (\(h\))**: The dimensions of \(h\) can be derived from the relation \(p \lambda = h\), where \(p\) is momentum and \(\lambda\) is wavelength.
- Momentum (\(p\)) has dimensions:
\[ [p] = [m][v] = M \cdot L \cdot T^{-1} \]
- Wavelength (\(\lambda\)) has dimensions:
\[ [\lambda] = L \]
- Therefore, the dimensions of \(h\) are:
\[ [h] = [p][\lambda] = (M \cdot L \cdot T^{-1}) \cdot L = M \cdot L^2 \cdot T^{-1} \]

3. **Mass of the electron (\(m_e\))**: The dimension of mass is simply:
\[ [m_e] = M \]

### Step 3: Substitute the dimensions into the formula for \(\mu_n\)
Now we can substitute the dimensions into the formula for \(\mu_n\):
\[ \mu_n = \frac{[e][h]}{[m_e]} \]
Substituting the dimensions we found:
\[ [\mu_n] = \frac{(A \cdot s)(M \cdot L^2 \cdot T^{-1})}{M} \]

### Step 4: Simplify the expression
When we simplify this expression:
\[ [\mu_n] = (A \cdot s)(L^2 \cdot T^{-1}) \]
The \(M\) in the numerator and denominator cancels out:
\[ [\mu_n] = A \cdot L^2 \cdot T^{-1} \]

### Final Result
Thus, the dimensions of the nuclear magneton \(\mu_n\) are:
\[ [\mu_n] = L^2 \cdot A \cdot T^{-1} \]