Problem 28 - Master Resource Book in JEE Main Physics

Question

A circular coil of radius \(8.0 \mathrm{~cm}\) and 20 turns is rotated about its vertical diameter with an angular speed of \(50 \mathrm{rad} / \mathrm{s}\) in a uniform horizontal magnetic field of magnitude \(3.0 \times 10^{-2} \mathrm{~T}\). Obtain the maximum and average emf induced in the coil. If the coil forms a closed-loop of resistance \(10 \Omega\), calculate the maximum value of current in the coil. Calculate the average power loss due to Joule heating. [NCERT] (a) \(2 \mathrm{~W}\) (b) \(0.2 \mathrm{~W}\) [c) \(0.49 \mathrm{~W}\) (d) \(0.018 \mathrm{~W}\)

Step-by-Step Solution

Answer

Maximum emf is 0.60 V; average emf is 0 V; max current is 0.06 A; power loss is 0.018 W.

Step 1: Understanding the Problem

The exercise involves calculating the maximum and average induced emf in a rotating coil, as well as the maximum current and power loss due to resistance. The coil rotates in a magnetic field, so Faraday's law of electromagnetic induction will be applied.

Step 2: Calculate Maximum Induced emf

The formula for the maximum emf (\( \varepsilon_{max} \)) induced in a coil is \( \varepsilon_{max} = NAB\omega \). Given \( N = 20 \) turns, \( A = \pi r^2 = \pi (0.08 \text{ m})^2 \), \( B = 3.0 \times 10^{-2} \text{ T} \), and \( \omega = 50 \text{ rad/s} \), calculate \( \varepsilon_{max} \).\[A = \pi \times (0.08)^2 = 0.020106 \text{ m}^2\]\[ \varepsilon_{max} = 20 \times 0.020106 \times 3.0 \times 10^{-2} \times 50 = 0.60 \text{ V} \]

Step 3: Determine Average Emf

The average emf over one complete rotation is zero because it is a sinusoidal waveform and averages out over a cycle. The average emf, in this case, is zero \( \varepsilon_{avg} = 0 \).

Step 4: Calculate Maximum Current in Coil

Using Ohm's Law, the maximum current \( I_{max} \) can be found using \( \varepsilon_{max} \) and resistance \( R = 10 \Omega \):\[ I_{max} = \frac{\varepsilon_{max}}{R} = \frac{0.60}{10} = 0.06 \text{ A} \]

Step 5: Calculate Average Power Loss due to Joule Heating

The average power loss \( P \) due to resistance in a coil is calculated using the equation \( P = \frac{1}{2} \varepsilon_{max} I_{max} \):\[P = \frac{1}{2} \times 0.60 \times 0.06 = 0.018 \text{ W}\]Thus, the average power loss due to Joule heating is \( 0.018 \text{ W} \).

Key Concepts

Faraday's Law

Faraday's Law of electromagnetic induction is a fundamental principle that describes how an electric field is generated by a changing magnetic field. This law states that the induced electromotive force (emf) in any closed circuit is equal to the rate of change of the magnetic flux through the circuit. In mathematical terms, Faraday's Law can be expressed as:
\[ ext{emf} = -N \frac{d ext{Φ}_m}{dt} \]where:

• \( N \): Number of turns in the coil
• \( ext{Φ}_m \): Magnetic flux
• \( \frac{d ext{Φ}_m}{dt} \): Rate of change of magnetic flux

In our exercise, Faraday's Law is used to calculate the maximum induced emf when the coil rotates at a certain speed in the magnetic field. The formula simplifies to \( ext{emf} = NAB\omega \) for a coil rotating uniformly, where:

• \( A \): Area of the coil, obtained from its radius
• \( B \): Magnetic field strength
• \( \omega \): Angular speed of rotation

By applying these principles correctly, we find that a maximum emf of 0.60 V is induced in the coil.

Joule Heating

Joule Heating, also known as resistive or ohmic heating, refers to the process where the electric current passing through a conductor produces thermal energy (heat) due to the resistance. The amount of heat produced can be determined by the formula:
\[P = I^2 R\]where:

• \( P \): Power (heat) produced
• \( I \): Current through the conductor
• \( R \): Resistance of the conductor

In the problem at hand, we calculate the average power loss due to Joule heating in the coil by using the maximum current and resistance. Since the current varies sinusoidally in this setup, the average power can also be determined using:
\[P_{avg} = \frac{1}{2} \varepsilon_{max} I_{max}\]In this instance, with a resistance of 10 Ω and a maximum current of 0.06 A, the average power loss due to Joule heating is found to be 0.018 W. This formula shows how resistance can convert electrical energy into heat, emphasizing the importance of resistor efficiency in electronic devices.

Ohm's Law

Ohm's Law is a fundamental principle that describes the relationship between voltage, current, and resistance in an electrical circuit. It is expressed by the equation:
\[V = IR\]where:

• \( V \): Voltage across the conductor
• \( I \): Current flowing through the conductor
• \( R \): Resistance of the conductor

In our context, Ohm's Law is instrumental in determining the maximum current (\( I_{max} \)) that flows through the coil when the maximum emf (\( \varepsilon_{max} \)) is induced. By rearranging the formula, we find that:
\[I_{max} = \frac{\varepsilon_{max}}{R}\]This calculation shows how voltage applied to a circuit and resistance within that circuit determines the current. For this exercise, with \( \varepsilon_{max} = 0.60 \, V \) and resistance \( R = 10 \, \Omega \), the resultant maximum current is 0.06 A. Understanding Ohm's Law helps in predicting how circuits will respond to changing electrical conditions.