Question

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

• A. $Q=\frac{1}{R}.\frac{Δ\phi}{Δt}$
• B. $Q=\frac{Δ\phi}{R}$
• C. $Q=\frac{Δ\phi}{Δt}$
• D. $Q=R.\frac{Δ\phi}{Δt}$

Hint:

When the magnetic flux linked with a circuit changes, an emf is induced in the circuit that stays as long as the flux keeps changing. 

Answer 1

The Correct Option is B

To solve the given problem, we need to find the total quantity of electric charge \( Q \) that passes through any point in the circuit during the time \( \Delta t \) when the magnetic flux \( \Delta \phi \) changes through a circuit of resistance \( R \).

According to Faraday’s law of electromagnetic induction, the electromotive force (EMF) induced in a circuit is given by:

\text{EMF} = -\frac{\Delta \phi}{\Delta t}

However, for the purpose of finding the charge, we are primarily interested in the magnitude of EMF.

Ohm’s Law states that EMF is also related to resistance and current by the formula:

\text{EMF} = I \cdot R

Where \( I \) is the current through the circuit. From the above formulas, we equate the expressions for EMF:

I \cdot R = \frac{\Delta \phi}{\Delta t}

Solving for the current \( I \), we get:

I = \frac{\Delta \phi}{R \cdot \Delta t}

The total charge \( Q \) that passes through any point in the circuit can be found by integrating the current over the time \( \Delta t \):

Q = I \cdot \Delta t = \frac{\Delta \phi}{R \cdot \Delta t} \cdot \Delta t

Simplifying this, we find:

Q = \frac{\Delta \phi}{R}

Hence, the correct answer is the second option, which is:

Q = \frac{\Delta \phi}{R}

This understanding aligns with the concept of charge induced by a change in magnetic flux, which is directly proportional to the change in flux and inversely proportional to the resistance of the circuit.