Dimensional formula of self-inductance is

Options

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

Explanation

To find the dimensional formula of self-inductance (L), we can start from the relationship between energy (E), inductance (L), and current (I). The formula we will use is:

\[ E = \frac{1}{2} L I^2 \]

### Step 1: Rearranging the formula
From the equation, we can express inductance (L) in terms of energy (E) and current (I):

\[ L = \frac{2E}{I^2} \]

### Step 2: Finding the dimensional formula for energy (E)
The dimensional formula for energy is derived from its relation to work. Work is defined as force multiplied by distance. The dimensional formula for force (F) is:

\[ F = M L T^{-2} \]

where:
- M = mass
- L = length
- T = time

Thus, the dimensional formula for energy (E) is:

\[ E = F \cdot d = (M L T^{-2}) \cdot L = M L^2 T^{-2} \]

### Step 3: Finding the dimensional formula for current (I)
The dimensional formula for electric current (I) is represented as:

\[ I = A \]

where A is the unit of electric current (amperes). Therefore, the dimensional formula for current is:

\[ [I] = A \]

### Step 4: Finding the dimensional formula for \( I^2 \)
Since we need \( I^2 \) for our equation, we square the dimensional formula for current:

\[ [I^2] = A^2 \]

### Step 5: Substituting the dimensional formulas into the equation for L
Now we can substitute the dimensional formulas for energy and current into the equation for inductance:

\[ [L] = \frac{[E]}{[I^2]} = \frac{M L^2 T^{-2}}{A^2} \]

### Step 6: Simplifying the expression
This gives us the dimensional formula for self-inductance:

\[ [L] = M L^2 T^{-2} A^{-2} \]

### Final Answer
Thus, the dimensional formula for self-inductance (L) is:

\[ [L] = M^1 L^2 T^{-2} A^{-2} \]

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