Problem 42 - Master Resource Book in JEE Main Physics

Question

A coil of inductance \(0.2 \mathrm{H}\) and \(1.0 \mathrm{~W}\) resistance is connected to a \(90 \mathrm{~V}\) source. At what rate will the current in the coil grow at the instant the coil is connected to the source? (a) \(450 \mathrm{As}^{-1}\) (b) \(4.5 \mathrm{As}^{-1}\) (c) \(45 \mathrm{As}^{-1}\) (d) \(0.45 \mathrm{As}^{-1}\)

Step-by-Step Solution

Answer

The current in the coil grows at a rate of 450 As^{-1}.

Step 1: Identify the Given Values

We are given:- Inductance \( L = 0.2 \text{ H} \)- Resistance \( R = 1.0 \underline{\phantom{xxx}} \Omega \)- Voltage \( V = 90 \text{ V} \).

Step 2: Use Ohm's Law to Find Initial Current

Ohm's Law states that \( V = IR \). Since the current is initially zero when the coil is just connected, use the voltage equation for LR circuits immediately after connection: \[ V = L \frac{di}{dt} + iR \]Initially, \( i = 0 \), so:\[ V = L \frac{di}{dt} \]\[ 90 = 0.2 \cdot \frac{di}{dt} \]

Step 3: Solve for \(\frac{di}{dt}\)

Rearrange the equation to find \( \frac{di}{dt} \):\[ \frac{di}{dt} = \frac{90}{0.2} \]Now, perform the division:\[ \frac{di}{dt} = 450 \text{ As}^{-1} \]

Step 4: Select the Answer

Based on our calculation, the rate at which the current grows is \( 450 \text{ As}^{-1} \). Looking at the options given, the correct answer is (a) \( 450 \text{ As}^{-1} \).

Key Concepts

Inductance

Inductance is a fundamental concept in electromagnetism. It describes a property of electrical circuits where they oppose changes in current. A common illustration of inductance is a coil or inductor.
Inductance is measured in henries (H), which quantifies how much voltage will be induced for a change in current. For instance, an inductance of 1 Henry means that a change in current of 1 ampere per second induces 1 volt across the coil.
Some key points to remember about inductors and inductance include:

• Inductors store energy in a magnetic field when electrical current flows through them.
• The induced voltage in an inductor is always opposite to the change in current due to Lenz's Law.
• Inductance depends on factors like the number of turns in the coil, the cross-sectional area, and the core material.

Understanding inductance helps in analyzing circuits, especially those involving changes in current, like LR circuits.

Ohm's Law

Ohm’s Law is one of the cornerstones of electrical engineering. It defines the relationship between voltage, current, and resistance in an electrical circuit. Mathematically, it is expressed as:\[ V = IR \]where:

• \( V \) is the voltage across the circuit in volts.
• \( I \) is the current flowing through the circuit in amperes.
• \( R \) is the resistance of the circuit in ohms (\( \Omega \)).

Ohm's Law implies that the current in a circuit is directly proportional to the voltage and inversely proportional to the resistance. This means:

• Increasing voltage increases the current, assuming resistance is constant.
• Increasing resistance decreases current, given a constant voltage.

This law is fundamental when analyzing simple circuits and is a stepping stone to tackling more complex scenarios like in LR circuits where the resistance and inductive reactance come into play.

LR Circuits

An LR circuit is a circuit containing both inductance (L) and resistance (R). This combination affects how circuits respond to changes in voltage and how quickly current changes over time.
An essential characteristic of LR circuits is their time constant, often denoted by \( \tau \), which is defined as:\[ \tau = \frac{L}{R} \]This time constant determines how fast the circuit responds to changes in voltage. Specifically, it influences the rate at which current either rises or decays:
- When the circuit is closed, initially the inductor resists changes in current. Therefore, the current takes some time to reach its maximum value, determined by these factors.- The voltage equation for LR circuits when first connected is \( V = L \frac{di}{dt} + iR \), where initially \( i = 0 \), making it possible to solve for the rate \( \frac{di}{dt} \).
- As shown in the original exercise, by isolating \( \frac{di}{dt} \), you can determine how rapidly the current increases right after the switch is closed, which is crucial in understanding how inductors and resistors interact in these systems.
Mastering LR circuits requires grasping both the temporal aspect and interaction of passive components like resistors and inductors, revealing how energy is stored and dissipated over time.