Using `mass (M), "length"(L) , time (T)` and current` (A) `as fundamental quantites the demension of permeability is

Options

A. `[M^(-1)LT^(-2)A]`
B. `[ML^(-2)T^(-2)A^(-1)]`
C. `[MLT^(-2)A^(-2)]`
D. `[MLT^(-1)A^(-1)]`

Explanation

To find the dimension of permeability using mass (M), length (L), time (T), and current (A) as fundamental quantities, we can follow these steps:

### Step 1: Understand the relationship involving permeability
We know that the speed of light \( c \) is related to permeability \( \mu_0 \) and permittivity \( \epsilon_0 \) by the equation:
\[ c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} \]
From this equation, we can express permeability as:
\[ \mu_0 = \frac{1}{c^2 \epsilon_0} \]

### Step 2: Determine the dimensions of speed of light \( c \)
The dimension of speed \( c \) is given by:
\[ [c] = [L^1 T^{-1}] \]

### Step 3: Find the dimensions of permittivity \( \epsilon_0 \)
Using the formula for the force between two charges, we have:
\[ F = \frac{1}{4\pi \epsilon_0} \frac{q_1 q_2}{r^2} \]
From this, we can rearrange to find \( \epsilon_0 \):
\[ \epsilon_0 = \frac{q_1 q_2}{F r^2} \]

### Step 4: Determine the dimensions of charge \( q \)
The dimension of electric charge \( q \) can be expressed in terms of current and time:
\[ [q] = [A^1 T^1] \]

### Step 5: Substitute the dimensions into the equation for \( \epsilon_0 \)
Now substituting the dimensions into the equation for \( \epsilon_0 \):
\[ [\epsilon_0] = \frac{[q]^2}{[F][r^2]} = \frac{(A^1 T^1)^2}{[M L T^{-2}][L^2]} = \frac{A^2 T^2}{M L^3} \]

### Step 6: Substitute \( \epsilon_0 \) into the equation for \( \mu_0 \)
Now substituting \( [\epsilon_0] \) back into the equation for \( \mu_0 \):
\[ [\mu_0] = \frac{1}{[c^2][\epsilon_0]} = \frac{1}{[L^2 T^{-2}][\frac{A^2 T^2}{M L^3}]} \]

### Step 7: Simplify the expression for \( \mu_0 \)
Now we simplify:
\[ [\mu_0] = \frac{1}{[L^2 T^{-2}] \cdot [\frac{A^2 T^2}{M L^3}]} = \frac{M L^3}{A^2 T^4} \]

### Step 8: Final expression for dimensions of permeability
Thus, the dimensions of permeability \( \mu_0 \) are:
\[ [\mu_0] = [M^1 L^1 T^{-2} A^{-2}] \]

### Conclusion
The dimension of permeability is:
\[ [M^1 L^1 T^{-2} A^{-2}] \]