A uniform magnetic flux density of 0.1 Wb/m extends over a plane circuit of area 2 m2 and is normal to it. How quickly must the field be reduced to zero, if an emf of 100 V is to be induced in the circuit?

Answer

To solve the problem, we need to calculate how quickly the magnetic field must be reduced to zero in order to induce an emf of 100 V in the circuit. We will use Faraday's law of electromagnetic induction, which states that the induced emf (E) in a closed loop is equal to the negative rate of change of magnetic flux (Φ) through the loop.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Magnetic flux density (B) = 0.1 Wb/m²
- Area of the circuit (A) = 2 m²
- Induced emf (E) = 100 V

2. **Calculate the Initial Magnetic Flux (Φ_initial):**
The magnetic flux (Φ) through the circuit is given by the formula:
\[ \Phi = B \cdot A \]
Substituting the values:
\[ \Phi_{\text{initial}} = 0.1 \, \text{Wb/m²} \times 2 \, \text{m²} = 0.2 \, \text{Wb} \]

3. **Determine the Final Magnetic Flux (Φ_final):**
Since we want to reduce the magnetic field to zero, the final magnetic flux will be:
\[ \Phi_{\text{final}} = 0 \, \text{Wb} \]

4. **Calculate the Change in Magnetic Flux (ΔΦ):**
The change in magnetic flux is:
\[ \Delta \Phi = \Phi_{\text{final}} - \Phi_{\text{initial}} = 0 - 0.2 = -0.2 \, \text{Wb} \]

5. **Apply Faraday's Law:**
According to Faraday's law:
\[ E = -\frac{d\Phi}{dt} \]
Rearranging gives:
\[ dt = -\frac{d\Phi}{E} \]
Substituting the values:
\[ dt = -\frac{-0.2 \, \text{Wb}}{100 \, \text{V}} = \frac{0.2}{100} = 0.002 \, \text{s} \]

6. **Convert Time to Milliseconds:**
\[ dt = 0.002 \, \text{s} = 2 \, \text{ms} \]

### Final Answer:
The magnetic field must be reduced to zero in **2 milliseconds** to induce an emf of 100 V in the circuit.