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

Options

A. `Q= (1)/(R)(Delta phi)/(Delta t)`
B. `Q= (Delta phi)/(R)`
C. `Q= (Delta phi)/(Delta t)`
D. `Q= R. (Delta phi)/(Delta t)`

Answer

To solve the problem, we need to determine the total quantity of electric charge \( Q \) that passes through any point in the circuit during the time interval \( \Delta t \), given that the magnetic flux \( \Delta \Phi \) through the circuit changes in that time.

### 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 equal to the negative rate of change of magnetic flux through the circuit:
\[ \mathcal{E} = -\frac{d\Phi}{dt} \]
In terms of the magnitude, we can express this as:
\[ \mathcal{E} = \frac{\Delta \Phi}{\Delta t} \]

2. **Applying Ohm's Law**:
The current \( I \) in the circuit can be calculated using Ohm's Law, which states that the current is equal to the EMF divided by the resistance \( R \):
\[ I = \frac{\mathcal{E}}{R} \]
Substituting the expression for EMF:
\[ I = \frac{\Delta \Phi / \Delta t}{R} = \frac{\Delta \Phi}{R \Delta t} \]

3. **Relating Current to Charge**:
The current \( I \) is also defined as the rate of flow of charge \( Q \) over time \( \Delta t \):
\[ I = \frac{dQ}{dt} \]
To find the total charge \( Q \) that flows during the time interval \( \Delta t \), we can rearrange this equation:
\[ Q = I \cdot \Delta t \]

4. **Substituting for Current**:
Now, substituting the expression for current \( I \) into the equation for charge:
\[ Q = \left(\frac{\Delta \Phi}{R \Delta t}\right) \cdot \Delta t \]
The \( \Delta t \) in the numerator and denominator cancels out:
\[ Q = \frac{\Delta \Phi}{R} \]

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

### Summary:
The final expression for the total charge \( Q \) is:
\[ Q = \frac{\Delta \Phi}{R} \]