If flux in a coil changes by `Deltaphi`, and the resistance of the coil is `R`,prove that the charge flown in the coil during the flux change is `(Deltaphi)/R`.(Note :It is independent of the time taken for the change is flux)

Explanation

To prove that the charge \( Q \) that flows in the coil during the change in flux \( \Delta \phi \) is given by the formula \( Q = \frac{\Delta \phi}{R} \), we will follow these steps:

### Step 1: Understand Faraday's Law of Electromagnetic Induction
According to Faraday's law, the electromotive force (emf) \( \mathcal{E} \) induced in a coil is equal to the negative rate of change of magnetic flux through the coil:
\[ \mathcal{E} = -\frac{d\phi}{dt} \]

### Step 2: Relate emf to Current
From Ohm's law, we know that the current \( I \) flowing through the coil can be expressed as:
\[ I = \frac{V}{R} \]
where \( V \) is the voltage (or emf in this case) and \( R \) is the resistance of the coil. Therefore, we can write:
\[ I = \frac{\mathcal{E}}{R} \]

### Step 3: Substitute emf into the Current Equation
Substituting the expression for emf from Step 1 into the equation for current gives:
\[ I = \frac{-\frac{d\phi}{dt}}{R} \]

### Step 4: Relate Current to Charge
Current is defined as the rate of flow of charge, which can be expressed as:
\[ I = \frac{dQ}{dt} \]
where \( Q \) is the charge. Therefore, we can equate the two expressions for current:
\[ \frac{dQ}{dt} = -\frac{1}{R} \frac{d\phi}{dt} \]

### Step 5: Integrate to Find Total Charge
To find the total charge \( Q \) that flows during the change in flux \( \Delta \phi \), we need to integrate both sides with respect to time. Assuming the change in flux occurs over a time interval \( \Delta t \):
\[ Q = -\int_0^{\Delta t} \frac{1}{R} \frac{d\phi}{dt} dt \]
Since \( \frac{d\phi}{dt} \) represents the change in flux, we can simplify this to:
\[ Q = -\frac{1}{R} \Delta \phi \]

### Step 6: Remove the Negative Sign
The negative sign indicates the direction of the induced emf and current. However, when calculating the magnitude of charge, we can express it as:
\[ Q = \frac{\Delta \phi}{R} \]

### Conclusion
Thus, we have shown that the charge \( Q \) that flows in the coil during the change in flux \( \Delta \phi \) is given by:
\[ Q = \frac{\Delta \phi}{R} \]
This result is independent of the time taken for the change in flux.

---