If the flux of magnetic induction through each turn of a coil of resistance R and having N turns changes from `phi_(1)` to `phi_(2)` then the magnitude of the charge that passes through the coil is

Options

A. `(phi_(2)-phi_(1))/(R)`
B. `(N(phi_(2)-phi_(1)))/(R)`
C. `(phi_(2)-phi_(1))/(NR)`
D. `(NR)/(phi_(2)-phi_(1))`

Answer

To find the magnitude of the charge that passes through a coil of resistance \( R \) with \( N \) turns when the magnetic flux changes from \( \phi_1 \) to \( \phi_2 \), we can 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. Mathematically, it is expressed as:
\[ \mathcal{E} = -N \frac{d\Phi}{dt} \]
where \( N \) is the number of turns and \( \Phi \) is the magnetic flux.

### Step 2: Calculate the Change in Flux
The change in magnetic flux \( \Delta \Phi \) through the coil is given by:
\[ \Delta \Phi = \Phi_2 - \Phi_1 \]

### Step 3: Relate emf to Current and Resistance
Using Ohm's law, we know that the current \( I \) flowing through the coil is related to the emf and the resistance \( R \) by:
\[ \mathcal{E} = I R \]
From this, we can express the current as:
\[ I = \frac{\mathcal{E}}{R} \]

### Step 4: Substitute emf in Terms of Change in Flux
Substituting the expression for emf from Faraday's law into the equation for current gives:
\[ I = \frac{N \frac{d\Phi}{dt}}{R} \]

### Step 5: Relate Charge to Current
The charge \( Q \) that passes through the coil in a time interval \( \Delta t \) can be expressed as:
\[ Q = I \Delta t \]
Substituting the expression for \( I \) into this equation gives:
\[ Q = \left(\frac{N \Delta \Phi}{R}\right) \Delta t \]

### Step 6: Final Expression for Charge
Since \( \Delta \Phi = \Phi_2 - \Phi_1 \), we can write:
\[ Q = \frac{N (\Phi_2 - \Phi_1)}{R} \Delta t \]

### Conclusion
Thus, the magnitude of the charge that passes through the coil is:
\[ Q = \frac{N (\Phi_2 - \Phi_1)}{R} \Delta t \]