The correct order in which the dimensions of 'Length" increases in the following physical quantities is
A) Permittivity
B) Resistance
C) Magnetic permeability
D) Stress

Options

A. A, B, C, D
B. D, C, B, A
C. A, D, C, B
D. C, B, D, A

Answer

To determine the correct order in which the dimensions of 'Length' increase for the given physical quantities, we will analyze the dimensions of each quantity step by step.

### Step-by-step Solution:

1. **Permittivity (ε)**:
- The formula for force between two charges is given by:
\[ F = \frac{1}{4\pi \epsilon} \frac{q_1 q_2}{r^2} \]
- Rearranging gives:
\[ \epsilon = \frac{q_1 q_2}{F r^2} \]
- The dimensions of force (F) are:
\[ [F] = M L T^{-2} \]
- The dimensions of charge (q) can be expressed in terms of current (I) and time (T):
\[ [q] = A T \]
- Therefore, the dimensions of \( q_1 q_2 \) are:
\[ [q_1 q_2] = (A T)^2 = A^2 T^2 \]
- Plugging these into the equation for ε gives:
\[ [\epsilon] = \frac{A^2 T^2}{(M L T^{-2}) L^2} = \frac{A^2 T^2}{M L^3 T^{-2}} = M^{-1} L^{-3} T^4 A^2 \]

2. **Resistance (R)**:
- The relation for resistance is given by:
\[ R = \frac{V}{I} \]
- The dimensions of voltage (V) are:
\[ [V] = M L^2 T^{-3} A^{-1} \]
- Thus, the dimensions of resistance are:
\[ [R] = \frac{M L^2 T^{-3} A^{-1}}{A} = M L^2 T^{-3} A^{-2} \]

3. **Magnetic Permeability (μ)**:
- The dimensions of magnetic permeability are given by:
\[ [\mu] = \frac{F}{A \cdot (T^2)} = \frac{M L T^{-2}}{L^2} = M L T^{-2} A^{-2} \]

4. **Stress (σ)**:
- Stress is defined as force per unit area:
\[ \sigma = \frac{F}{A} \]
- Therefore, the dimensions of stress are:
\[ [\sigma] = \frac{M L T^{-2}}{L^2} = M L^{-1} T^{-2} \]

### Summary of Dimensions:
- **Permittivity (ε)**: \( M^{-1} L^{-3} T^4 A^2 \)
- **Resistance (R)**: \( M L^2 T^{-3} A^{-2} \)
- **Magnetic Permeability (μ)**: \( M L T^{-2} A^{-2} \)
- **Stress (σ)**: \( M L^{-1} T^{-2} \)

### Ordering by Length Dimensions:
- For **Permittivity**: \( L^{-3} \)
- For **Stress**: \( L^{-1} \)
- For **Magnetic Permeability**: \( L^{1} \)
- For **Resistance**: \( L^{2} \)

Thus, the correct order in which the dimensions of 'Length' increase is:
1. Permittivity (L^-3)
2. Stress (L^-1)
3. Magnetic Permeability (L^1)
4. Resistance (L^2)

### Final Answer:
The correct order is: **A (Permittivity), D (Stress), C (Magnetic Permeability), B (Resistance)**.