The dimensional formula for induced emf?

Options

A. `ML^3T^(-1)A^(-1)`
B. `MLT^(-2)A^(-1)`
C. `ML^2T^(-3)A^(-1)`
D. `MLT^(-3)A^(-1)`

Explanation

To find the dimensional formula for induced electromotive force (emf), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Relationship**:
Induced emf (E) is related to work done (W) and charge (Q) by the formula:
\[ E = \frac{W}{Q} \]

2. **Identify the Dimensions**:
We need to find the dimensions of work (W) and charge (Q).

3. **Dimension of Work**:
Work is defined as the product of force and displacement. The formula for work is:
\[ W = F \cdot d \]
where \(F\) is force and \(d\) is displacement.

- The dimensional formula for force (F) is given by:
\[ F = m \cdot a = m \cdot \frac{L}{T^2} \quad \text{(where } m \text{ is mass, } a \text{ is acceleration)} \]
Thus, the dimensional formula for force is:
\[ [F] = M^1 L^1 T^{-2} \]

- The dimensional formula for displacement (d) is:
\[ [d] = L^1 \]

- Therefore, the dimensional formula for work (W) becomes:
\[ [W] = [F] \cdot [d] = (M^1 L^1 T^{-2}) \cdot (L^1) = M^1 L^2 T^{-2} \]

4. **Dimension of Charge**:
Charge (Q) can be expressed in terms of current (I) and time (t):
\[ Q = I \cdot t \]
- The dimensional formula for current (I) is:
\[ [I] = A^1 \]
- The dimensional formula for time (t) is:
\[ [t] = T^1 \]
- Therefore, the dimensional formula for charge (Q) becomes:
\[ [Q] = [I] \cdot [t] = (A^1) \cdot (T^1) = A^1 T^1 \]

5. **Combine the Dimensions**:
Now, substituting the dimensions of work and charge back into the equation for induced emf:
\[ [E] = \frac{[W]}{[Q]} = \frac{M^1 L^2 T^{-2}}{A^1 T^1} \]

6. **Simplify the Expression**:
Simplifying the above expression:
\[ [E] = M^1 L^2 T^{-2} \cdot A^{-1} T^{-1} = M^1 L^2 A^{-1} T^{-3} \]

7. **Final Result**:
The dimensional formula for induced emf is:
\[ [E] = M^1 L^2 A^{-1} T^{-3} \]