The quantity `[(nh)//(2piqB)]^(1//2)` where `n` is a positive integer, `h` is Planck's constant `q` is charge and `B` is magnetic field has the dimensions of

Options

A. area
B. speed
C. length
D. acceleration

Answer

To find the dimensions of the quantity \(\left[\frac{nh}{2\pi qB}\right]^{1/2}\), we will analyze the dimensions of each component in the expression.

### Step-by-Step Solution:

1. **Identify the constants and their dimensions**:
- \(h\) (Planck's constant) has dimensions of energy multiplied by time. In terms of fundamental dimensions, it can be expressed as:
\[ [h] = [E][T] = [M][L^2][T^{-2}][T] = [M][L^2][T^{-1}] \]
- \(q\) (charge) has dimensions of electric charge, which is typically represented as:
\[ [q] = [I][T] \]
- \(B\) (magnetic field) has dimensions of magnetic flux density, which can be expressed in terms of force and charge:
\[ [B] = \frac{[F]}{[q][v]} = \frac{[M][L][T^{-2}]}{[I][T][L]} = [M][T^{-2}][I^{-1}] \]

2. **Substituting the dimensions into the expression**:
- Now, we substitute the dimensions into the expression \(\frac{nh}{2\pi qB}\):
\[ \frac{h}{qB} = \frac{[M][L^2][T^{-1}]}{[I][T] \cdot [M][T^{-2}][I^{-1}]} \]
- Simplifying this expression:
\[ = \frac{[M][L^2][T^{-1}]}{[M][T^{-2}][I]} = \frac{[L^2][T^{-1}]}{[T^{-2}][I]} = [L^2][T][I^{-1}] \]

3. **Taking the square root**:
- Now we take the square root of the entire expression:
\[ \left[\frac{nh}{2\pi qB}\right]^{1/2} = \left[L^2 T I^{-1}\right]^{1/2} = [L][T^{1/2}][I^{-1/2}] \]

4. **Final dimensions**:
- Therefore, the dimensions of the quantity \(\left[\frac{nh}{2\pi qB}\right]^{1/2}\) are:
\[ [L][T^{1/2}][I^{-1/2}] \]

### Conclusion:
The quantity \(\left[\frac{nh}{2\pi qB}\right]^{1/2}\) has dimensions of \([L][T^{1/2}][I^{-1/2}]\).