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 determine the total quantity of electric charge \( Q \) that passes through a circuit of resistance \( R \) when the magnetic flux changes by an amount \( \Delta \phi \) in a time \( \Delta t \).

### Step-by-Step Solution:

1. **Understanding Faraday's Law of Electromagnetic Induction**:
According to Faraday's law, the induced electromotive force (emf) \( \mathcal{E} \) in a circuit is given by the rate of change of magnetic flux through the circuit:
\[ \mathcal{E} = -\frac{\Delta \phi}{\Delta t} \]
Here, \( \Delta \phi \) is the change in magnetic flux and \( \Delta t \) is the time over which this change occurs.

2. **Relating Induced emf to Current**:
The induced emf causes a current \( I \) to flow through the circuit. According to Ohm's law, the current can be expressed as:
\[ I = \frac{\mathcal{E}}{R} \]
where \( R \) is the resistance of the circuit.

3. **Substituting for Induced emf**:
By substituting the expression for \( \mathcal{E} \) from Faraday's law into the equation for current, we get:
\[ I = \frac{-\Delta \phi / \Delta t}{R} = -\frac{\Delta \phi}{R \Delta t} \]

4. **Relating Current to Charge**:
The current \( I \) is also defined as the rate of flow of charge. Therefore, we can express the charge \( Q \) that passes through a point in the circuit during the time \( \Delta t \) as:
\[ Q = I \cdot \Delta t \]

5. **Substituting for Current**:
Now substituting the expression for \( I \) into the equation for \( Q \):
\[ Q = \left(-\frac{\Delta \phi}{R \Delta t}\right) \cdot \Delta t \]

6. **Simplifying the Expression**:
The \( \Delta t \) terms cancel out:
\[ Q = -\frac{\Delta \phi}{R} \]
Since we are interested in the magnitude of charge, we can ignore the negative sign:
\[ Q = \frac{\Delta \phi}{R} \]

### Final Answer:
Thus, the total quantity of electric charge \( Q \) that passes through any point in the circuit during the time \( \Delta t \) is given by:
\[ Q = \frac{\Delta \phi}{R} \]