What are the dimensions of electrical permittivity?

Options

A. `ML^(-2)T^(-2)Q^(-2)`
B. `M^(-1)L^(2)T^(-3)Q^(-1)`
C. `M^(-1)L^(-3)T^(2)Q^(2)`
D. `M^(-1)L^(3)T^(-2)Q^(-2)`

Explanation

To find the dimensions of electrical permittivity (ε), we can start from the formula for the force between two point charges. The formula is given by:

\[ F = \frac{1}{4\pi \epsilon} \cdot \frac{q_1 q_2}{r^2} \]

Where:
- \( F \) is the force between the charges,
- \( \epsilon \) is the electrical permittivity,
- \( q_1 \) and \( q_2 \) are the magnitudes of the charges,
- \( r \) is the distance between the charges.

### Step 1: Rearranging the formula
From the formula, we can express \( \epsilon \) in terms of the other quantities:

\[ \epsilon = \frac{q_1 q_2}{4\pi F r^2} \]

Since \( 4\pi \) is a constant, we can ignore its dimensions.

### Step 2: Finding the dimensions of each quantity
1. **Charge (q)**: The dimension of charge is denoted as \( [q] \).
2. **Force (F)**: The dimension of force is given by \( [F] = M L T^{-2} \).
3. **Distance (r)**: The dimension of distance is \( [r] = L \).

### Step 3: Substituting dimensions into the equation
Now we can substitute these dimensions into our expression for \( \epsilon \):

\[ [\epsilon] = \frac{[q]^2}{[F] \cdot [r]^2} \]

Substituting the dimensions we have:

\[ [\epsilon] = \frac{[q]^2}{(M L T^{-2}) \cdot (L^2)} \]

### Step 4: Simplifying the expression
This simplifies to:

\[ [\epsilon] = \frac{[q]^2}{M L^3 T^{-2}} \]

Rearranging gives:

\[ [\epsilon] = M^{-1} L^{-3} T^{2} [q]^2 \]

### Final Result
Thus, the dimensions of electrical permittivity \( \epsilon \) are:

\[ [\epsilon] = M^{-1} L^{-3} T^{2} [q]^2 \]