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 \(\left[\frac{nh}{2\pi q B}\right]^{1/2}\), we will analyze each component in the expression step by step.

### Step 1: Identify the dimensions of each component
1. **Planck's constant \(h\)**:
- The dimensions of Planck's constant \(h\) are given by:
\[ [h] = [E][T] = [M][L^2][T^{-1}] \]
- Here, \(E\) (energy) has the dimensions of \([M][L^2][T^{-2}]\), so:
\[ [h] = [M][L^2][T^{-1}] \]

2. **Charge \(q\)**:
- The dimensions of electric charge \(q\) are:
\[ [q] = [I][T] \]
- Here, \(I\) is the dimension of electric current.

3. **Magnetic field \(B\)**:
- The dimensions of the magnetic field \(B\) can be expressed as:
\[ [B] = \frac{[F]}{[q][v]} = \frac{[M][L][T^{-2}]}{[I][T][L/T]} = [M][T^{-2}][I^{-1}] \]

### Step 2: Substitute the dimensions into the expression
Now we can substitute the dimensions into the expression \(\left[\frac{nh}{2\pi q B}\right]^{1/2}\):

- The dimension of \(nh\) is the same as that of \(h\) since \(n\) is a dimensionless positive integer:
\[ [nh] = [h] = [M][L^2][T^{-1}] \]

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

- The dimension of \(qB\) is:
\[ [qB] = [q][B] = [I][T][M][T^{-2}][I^{-1}] = [M][T^{-1}] \]

### Step 3: Combine the dimensions
Now we can write the dimensions of the entire expression:
\[ \left[\frac{nh}{qB}\right] = \left[\frac{[M][L^2][T^{-1}]}{[M][T^{-1}]}\right] = [L^2] \]

### Step 4: Take the square root
Finally, we take the square root of the dimensions:
\[ \left[\frac{nh}{qB}\right]^{1/2} = [L^2]^{1/2} = [L] \]

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