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. length
C. speed
D. acceleration

Explanation

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

### Step 1: Identify the dimensions of each variable

1. **Planck's constant \(h\)**:
- The dimensions of Planck's constant \(h\) are given by:
\[ [h] = [E][T] = [M][L^2][T^{-1}] \]
- Where \(E\) is energy, which has dimensions \([M][L^2][T^{-2}]\).

2. **Charge \(q\)**:
- The dimensions of electric charge \(q\) are:
\[ [q] = [I][T] \]
- Where \(I\) is current and \(T\) is time.

3. **Magnetic field \(B\)**:
- The dimensions of the magnetic field \(B\) can be derived from the Lorentz force equation \(F = qvB\). Rearranging gives:
\[ [B] = \frac{[F]}{[q][v]} = \frac{[M][L][T^{-2}]}{[I][T][L/T]} = \frac{[M]}{[I][T]} = [M][I^{-1}][T^{-2}] \]

### Step 2: Substitute the dimensions into the expression

Now, substituting the dimensions into the expression \(\frac{nh}{2\pi qB}\):

- The term \(nh\) has dimensions:
\[ [nh] = [h] = [M][L^2][T^{-1}] \]

- The term \(2\pi\) is dimensionless, so it does not affect the dimensions.

- The term \(qB\) has dimensions:
\[ [qB] = [q][B] = ([I][T])([M][I^{-1}][T^{-2}]) = [M][T^{-1}] \]

### Step 3: Combine the dimensions

Now, we can combine the dimensions:
\[ \frac{nh}{qB} = \frac{[M][L^2][T^{-1}]}{[M][T^{-1}]} = [L^2] \]

### Step 4: Take the square root

Finally, we take the square root of the combined dimensions:
\[ \left[\frac{nh}{qB}\right]^{1/2} = [L^2]^{1/2} = [L] \]

### Conclusion

Thus, the dimensions of the quantity \(\left[\frac{nh}{2\pi qB}\right]^{1/2}\) are:
\[ \text{Dimensions} = [L] \]