The magnetic flux through a circuit of resistance `R` changes by an amount `Deltaphi` in a time `Deltat`. Then the total quantity of electric charge `Q` that passes any point in the circuit during the time `Deltat` is represent by

Options

A. `Q=(1)/(r ).(Deltaphi)/(Deltat)`
B. `Q=(Deltaphi)/(R )`
C. `Q=(Deltaphi)/(Deltat)`
D. `Q=(R ).(Deltaphi)/(Deltat)`

Explanation

To solve the problem, we need to apply Faraday's law of electromagnetic induction and Ohm's law.

### Step-by-step Solution:

1. **Understanding Faraday's Law**:
Faraday's law states that the induced electromotive force (emf) in a circuit is equal to the negative rate of change of magnetic flux through the circuit. Mathematically, it can be expressed as:
\[ \text{emf} = -\frac{d\Phi}{dt} \]
where \( \Phi \) is the magnetic flux.

2. **Change in Magnetic Flux**:
If the magnetic flux changes by an amount \( \Delta \Phi \) in a time interval \( \Delta t \), the average induced emf (\( \text{emf} \)) can be expressed as:
\[ \text{emf} = -\frac{\Delta \Phi}{\Delta t} \]

3. **Applying Ohm's Law**:
According to Ohm's law, the current \( I \) flowing through a circuit is related to the induced emf and the resistance \( R \) of the circuit:
\[ I = \frac{\text{emf}}{R} \]

4. **Substituting emf into Ohm's Law**:
By substituting the expression for emf from Faraday's law into Ohm's law, we get:
\[ I = \frac{-\Delta \Phi / \Delta t}{R} = -\frac{\Delta \Phi}{R \Delta t} \]

5. **Finding the Total Charge \( Q \)**:
The total charge \( Q \) that passes through any point in the circuit during the time interval \( \Delta t \) can be calculated using the relationship between current and charge:
\[ Q = I \cdot \Delta t \]
Substituting the expression for current \( I \):
\[ Q = \left(-\frac{\Delta \Phi}{R \Delta t}\right) \cdot \Delta t = -\frac{\Delta \Phi}{R} \]

6. **Final Expression**:
Thus, the total quantity of electric charge \( Q \) that passes any point in the circuit during the time \( \Delta t \) is given by:
\[ Q = -\frac{\Delta \Phi}{R} \]