What is the dimensional formula for the coefficient of self induction?

Options

A. `[L]=[M^(1)L^(2)T^(-2)]`
B. `[L]=[M^(1)L^(2)T^(-3)A^(1)]`
C. `[L]=[M^(1)L^(2)T^(-2)A^(-2)]`
D. `[L]=[M^(0)L^(1)T^(-2)A^(-3)]`

Explanation

To find the dimensional formula for the coefficient of self-induction (L), we start with the relationship between electromotive force (EMF), self-inductance, and the rate of change of current. The formula we use is:

\[ \text{EMF} = L \frac{di}{dt} \]

### Step 1: Rearranging the formula
From the above equation, we can express L (the coefficient of self-inductance) as:

\[ L = \frac{\text{EMF}}{\frac{di}{dt}} \]

### Step 2: Expressing EMF in terms of work and charge
We know that EMF can be expressed as work done per unit charge. Therefore, we can write:

\[ \text{EMF} = \frac{W}{Q} \]

Where:
- W = Work done (in joules)
- Q = Charge (in coulombs)

### Step 3: Substituting EMF into the equation for L
Substituting the expression for EMF into the equation for L, we get:

\[ L = \frac{W}{Q} \cdot \frac{dt}{di} \]

### Step 4: Finding the dimensions of work and charge
1. **Dimensions of Work (W)**:
Work is defined as force multiplied by displacement. The dimension of force (F) is given by:

\[ F = \text{mass} \times \text{acceleration} = M L T^{-2} \]

Therefore, the dimension of work (W) is:

\[ W = F \times \text{displacement} = (M L T^{-2}) \times L = M L^2 T^{-2} \]

2. **Dimensions of Charge (Q)**:
Charge is defined in terms of current and time. The dimension of current (I) is A (amperes), and time (t) is T. Thus, the dimension of charge (Q) is:

\[ Q = I \times t = A \times T \]

### Step 5: Substituting dimensions into the equation for L
Now we substitute the dimensions of W and Q into the equation for L:

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

### Step 6: Simplifying the expression
Now we simplify the expression for L:

\[ L = \frac{M L^2 T^{-2}}{A T} = M L^2 T^{-3} A^{-1} \]

### Final Result
Thus, the dimensional formula for the coefficient of self-induction (L) is:

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