The formula for induced e.m.f. in a coil due to change in magnetic flux through the coil is (here A = area of the coil, B = magnetic field)

Options

A. `e =-A.(dB)/(dt)`
B. `e =-B.(dA)/(dt)`
C. `e =-(d)/(dt)(A.B)`
D. `e =-(d)/(dt)(A xx B)`

Answer

To derive the formula for the induced electromotive force (e.m.f.) in a coil due to a change in magnetic flux, we can follow these steps:

### Step-by-Step Solution:

1. **Understand Faraday's Law of Electromagnetic Induction**:
According to Faraday's Law, the induced e.m.f. (E) in a coil is proportional to the rate of change of magnetic flux (Φ) through the coil. Mathematically, this is expressed as:
\[ E = -\frac{d\Phi}{dt} \]
where \(E\) is the induced e.m.f., and \(\Phi\) is the magnetic flux.

**Hint**: Recall that the negative sign indicates the direction of the induced e.m.f. opposes the change in flux (Lenz's Law).

2. **Define Magnetic Flux**:
The magnetic flux (Φ) through a surface is given by the dot product of the magnetic field (B) and the area vector (A) of the surface:
\[ \Phi = \mathbf{B} \cdot \mathbf{A} \]
where \(\mathbf{B}\) is the magnetic field vector and \(\mathbf{A}\) is the area vector.

**Hint**: Remember that the area vector is perpendicular to the surface and its magnitude is equal to the area of the surface.

3. **Substitute Magnetic Flux into Faraday's Law**:
Substitute the expression for magnetic flux into Faraday's Law:
\[ E = -\frac{d}{dt}(\mathbf{B} \cdot \mathbf{A}) \]

**Hint**: Keep in mind that the area vector may change if the area of the coil changes or if the orientation of the coil changes with respect to the magnetic field.

4. **Apply the Product Rule**:
If both the magnetic field and the area vector can change with time, we can apply the product rule of differentiation:
\[ E = -\left( \frac{d\mathbf{B}}{dt} \cdot \mathbf{A} + \mathbf{B} \cdot \frac{d\mathbf{A}}{dt} \right) \]

**Hint**: This step is crucial if the magnetic field strength or the orientation of the coil changes over time.

5. **Final Expression**:
The final expression for the induced e.m.f. can be summarized as:
\[ E = -\left( \frac{d\Phi}{dt} \right) = -\left( \frac{d}{dt}(\mathbf{B} \cdot \mathbf{A}) \right) \]

**Hint**: This formula indicates that the induced e.m.f. depends on how quickly the magnetic flux through the coil is changing.

### Conclusion:
The formula for the induced e.m.f. in a coil due to a change in magnetic flux is:
\[ E = -\frac{d\Phi}{dt} \]
where \(\Phi = \mathbf{B} \cdot \mathbf{A}\).