Which one of the following is not the dimensionless quantity?

Options

A. Plank's constant
B. Dielectric constant
C. Solid engle
D. Strain

Explanation

To determine which of the following quantities is not dimensionless, we need to analyze the dimensions of each option provided.

### Step-by-Step Solution:

1. **Identify the Options**: We need to evaluate the dimensions of the following quantities:
- Option A: Planck's constant
- Option B: Dielectric constant
- Option C: Solid angle
- Option D: Strain

2. **Evaluate Planck's Constant**:
- Planck's constant (h) is used in the equation for de Broglie wavelength:
\[ \lambda = \frac{h}{p} \]
where \( p \) is momentum.
- Rearranging gives:
\[ h = \lambda \cdot p \]
- The dimensions of momentum \( p \) are \( [M][L][T^{-1}] \) (mass times velocity).
- The dimension of wavelength \( \lambda \) is \( [L] \).
- Therefore, the dimensions of Planck's constant are:
\[ [h] = [L] \cdot [M][L][T^{-1}] = [M][L^2][T^{-1}] \]
- Since Planck's constant has dimensions, it is **not dimensionless**.

3. **Evaluate Dielectric Constant**:
- The dielectric constant is defined as the ratio of the permittivity of a substance to the permittivity of free space.
- Since it is a ratio of two similar quantities, it is dimensionless.

4. **Evaluate Solid Angle**:
- Solid angle is defined as the ratio of the area on a sphere to the square of the radius of that sphere:
\[ \text{Solid Angle} = \frac{dA}{R^2} \]
- Here, both the area \( dA \) and \( R^2 \) have dimensions of \( [L^2] \), so:
\[ \text{Solid Angle} = \frac{[L^2]}{[L^2]} = 1 \]
- Therefore, solid angle is also dimensionless.

5. **Evaluate Strain**:
- Strain is defined as the change in length divided by the original length:
\[ \text{Strain} = \frac{\Delta L}{L} \]
- Both \( \Delta L \) and \( L \) have dimensions of \( [L] \), so:
\[ \text{Strain} = \frac{[L]}{[L]} = 1 \]
- Hence, strain is dimensionless.

### Conclusion:
The only quantity that is not dimensionless among the options is **Planck's constant**.

### Final Answer:
**Option A: Planck's constant is not a dimensionless quantity.**

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