A current coil of total resistance R is kept inside magnetic field. How is the change in magnetic flux through the coil related to the amount of charge passing through the coil?

Answer

To find the relationship between the change in magnetic flux through a coil and the amount of charge passing through the coil, we can follow these steps:

### Step 1: Understand the relationship between EMF and magnetic flux
The electromotive force (EMF) induced in the coil due to a change in magnetic flux is given by Faraday's law of electromagnetic induction:
\[ E = -\frac{d\Phi}{dt} \]
where \(E\) is the EMF, \(\Phi\) is the magnetic flux, and \(t\) is time.

### Step 2: Relate EMF to current and resistance
According to Ohm's law, the relationship between EMF, current (\(I\)), and resistance (\(R\)) is given by:
\[ E = I \cdot R \]

### Step 3: Express current in terms of charge
Current can also be defined as the rate of flow of charge:
\[ I = \frac{dQ}{dt} \]
where \(Q\) is the charge.

### Step 4: Substitute current into the EMF equation
From the previous steps, we can substitute \(I\) into the EMF equation:
\[ E = \frac{dQ}{dt} \cdot R \]

### Step 5: Combine the equations
Now, we can equate the two expressions for EMF:
\[ -\frac{d\Phi}{dt} = \frac{dQ}{dt} \cdot R \]

### Step 6: Rearrange to find the relationship between charge and magnetic flux
Rearranging the equation gives:
\[ dQ = -\frac{d\Phi}{dt} \cdot R \cdot dt \]
This implies:
\[ dQ = -R \cdot \frac{d\Phi}{dt} \cdot dt \]

### Step 7: Simplifying the relationship
If we consider a small time interval, we can express the total charge \(Q\) that has passed through the coil as:
\[ Q = -R \cdot \Delta\Phi \]
where \(\Delta\Phi\) is the change in magnetic flux over the time interval.

### Final Relationship
Thus, the relationship between the change in magnetic flux through the coil and the amount of charge passing through the coil is:
\[ \Delta Q = -R \cdot \Delta \Phi \]