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)`

Answer

To find the dimensions of electrical permittivity (ε₀), we start with the relationship given by Coulomb's law, which states that the electrostatic force (F) between two charges (q₁ and q₂) is given by:

\[ F = \frac{1}{4\pi \epsilon_0} \cdot \frac{q_1 \cdot q_2}{r^2} \]

Where:
- \( F \) is the electrostatic force,
- \( \epsilon_0 \) is the electrical permittivity,
- \( q_1 \) and \( q_2 \) are the magnitudes of the charges,
- \( r \) is the distance between the charges.

### Step 1: Write down the dimensions of the electrostatic force.
The dimensions of force (F) can be expressed as:
\[ [F] = [M][L][T^{-2}] \]
Where:
- \( M \) is mass,
- \( L \) is length,
- \( T \) is time.

### Step 2: Write down the dimensions of charge.
The dimensions of charge (Q) can be represented as:
\[ [Q] = [Q] \]
We will keep it as \( [Q] \) for now since we need to derive its dimensions.

### Step 3: Substitute the dimensions into Coulomb's law.
From Coulomb's law, we can rearrange it to express ε₀:
\[ \epsilon_0 = \frac{q_1 \cdot q_2}{F \cdot r^2} \]

### Step 4: Substitute the dimensions into the equation.
Substituting the dimensions into the equation, we have:
\[ [\epsilon_0] = \frac{[Q]^2}{[M][L][T^{-2}] \cdot [L^2]} \]

### Step 5: Simplify the dimensions.
This simplifies to:
\[ [\epsilon_0] = \frac{[Q]^2}{[M][L^3][T^{-2}]} \]

### Step 6: Rearranging the expression.
Rearranging gives us:
\[ [\epsilon_0] = [M^{-1}][L^{-3}][T^2][Q^2] \]

### Conclusion:
Thus, the dimensions of electrical permittivity (ε₀) are:
\[ [\epsilon_0] = [M^{-1}][L^{-3}][T^2][Q^2] \]