`(he)/(4 pi m)` has the same dimensions as (h = Planck's constant , e = charge , m = mass.)

Options

A. Magnetic moment
B. Magnetic induction
C. Angular moment
D. Pole strength

Answer

To solve the problem where we need to find the dimensions of the expression \((he)/(4 \pi m)\), we will break it down step by step.

### Step 1: Identify the dimensions of each component
1. **Planck's constant (h)**: The dimension of Planck's constant is given by the formula:
\[ [h] = [Energy] \times [Time] = [M L^2 T^{-2}] \times [T] = [M L^2 T^{-1}] \]

2. **Charge (e)**: The dimension of charge can be expressed as:
\[ [e] = [Current] \times [Time] = [A] \times [T] \]
where \(A\) is the dimension of electric current.

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

### Step 2: Combine the dimensions
Now we can substitute the dimensions into the expression \((he)/(4 \pi m)\):
\[ \frac{he}{4 \pi m} = \frac{[h] \cdot [e]}{[m]} \]

Substituting the dimensions we found:
\[ = \frac{[M L^2 T^{-1}] \cdot [A T]}{[M]} \]

### Step 3: Simplify the expression
Now we simplify the expression:
\[ = \frac{[M L^2 T^{-1} A T]}{[M]} = [L^2 T^{0} A] = [L^2 A] \]

### Step 4: Identify the physical quantity
The dimension \([L^2 A]\) corresponds to the dimension of magnetic moment. Therefore, the expression \((he)/(4 \pi m)\) has the same dimensions as magnetic moment.

### Conclusion
Thus, the answer to the question is that \((he)/(4 \pi m)\) has the same dimensions as **magnetic moment**.

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