AIPMT PRELIMS 2004 Physics Faraday's Law MCQ Question

To understand this question, we need to delve into Faraday's Law of Electromagnetic Induction and the relationship between magnetic flux, induced electromotive force (emf), and electric charge.

Explanation of Faraday's Law

Faraday's Law states that the induced electromotive force ($\mathcal{E}$) in a closed loop is directly proportional to the rate of change of magnetic flux ($\Delta \Phi$) through the loop. Mathematically, it can be expressed as:

$$\mathcal{E} = -\frac{d\Phi}{dt}$$

For a change in magnetic flux $\Delta \Phi$ over a time interval $\Delta t$, we can approximate this as:

$$\mathcal{E} = -\frac{\Delta \Phi}{\Delta t}$$

Relationship Between Induced Emf, Resistance, and Charge

According to Ohm's Law, the relationship between current ($I$), voltage ($V$), and resistance ($R$) is given by:

$$I = \frac{V}{R}$$

In our case, the induced emf acts as the voltage across the circuit; thus, we can substitute $V$ with $\mathcal{E}$:

$$I = \frac{\mathcal{E}}{R} = \frac{-\Delta \Phi / \Delta t}{R} = -\frac{\Delta \Phi}{R \Delta t}$$

The current ($I$) can also be defined as the total charge ($Q$) passing through a point in the circuit over time $\Delta t$:

$$I = \frac{Q}{\Delta t}$$

Setting the two equations for current equal to each other gives us:

$$\frac{Q}{\Delta t} = -\frac{\Delta \Phi}{R \Delta t}$$

Multiplying both sides by $\Delta t$ leads to:

Q = -\frac{\Delta \Phi}{R} \quad \text{(The negative sign indicates direction, but we consider magnitude here.)}$$ ### Final Result This leads us to conclude that the total quantity of electric charge $ Q $ that passes any point in the circuit during the time $ \Delta t $ is given by:

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

Thus, the correct answer is **A) $ Q = \frac{\Delta \Phi}{R} $**. ### Clarification of Incorrect Options Let's briefly analyze the other options to clarify why they are incorrect: - **B) $ Q = \frac{\Delta \Phi}{\Delta t} $**: This expression suggests that the charge is directly proportional to the change in magnetic flux divided by the time interval. However, this does not account for the resistance in the circuit, which is essential for calculating the charge. - **C) $ Q = R \cdot \frac{\Delta \Phi}{\Delta t} $**: This option incorrectly implies that the charge is proportional to both resistance and the rate of change of magnetic flux. This is not accurate, as the charge is inversely proportional to resistance, as shown in our derivation. - **D) $ Q = \frac{1}{R} \cdot \frac{\Delta \Phi}{\Delta t} $**: Similar to option C, this expression incorrectly suggests that charge is proportional to the rate of change of magnetic flux divided by resistance. This again contradicts the principles of electromagnetic induction and Ohm's Law. ### Conclusion From our analysis, the correct formula relating the quantity of electric charge $ Q $ that passes through the circuit during the time $ \Delta t $ is indeed:

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

$$This understanding is critical in grasping the principles of electromagnetic induction and its applications in circuits.$$