FIGURE P30.48 shows two 20-turn coils tightly wrapped on the same 2.0-cm-diameter cylinder with 1.0-mm-diameter wire. The current through coil 1 is shown in the graph. Determine the current in coil 2 at (a) t = 0.05 s and (b) t = 0.25 s. A positive current is into the figure at the top of a loop. Assume that the magnetic field of coil 1 passes entirely through coil 2.

Verified step by step guidance

1. Step 1: Understand the problem. The two coils are tightly wrapped on the same cylinder, and the current in coil 1 changes over time, as shown in the graph. This changing current generates a changing magnetic field, which induces an electromotive force (EMF) in coil 2 according to Faraday's Law of Induction.
2. Step 2: Apply Faraday's Law of Induction. Faraday's Law states that the induced EMF in coil 2 is given by \( \text{EMF} = -N \frac{d\Phi_B}{dt} \), where \( N \) is the number of turns in coil 2, and \( \Phi_B \) is the magnetic flux through coil 2. The negative sign indicates the direction of the induced EMF opposes the change in flux.
3. Step 3: Calculate the magnetic flux \( \Phi_B \). The magnetic flux is given by \( \Phi_B = B \cdot A \), where \( B \) is the magnetic field produced by coil 1, and \( A \) is the cross-sectional area of the cylinder. The magnetic field \( B \) can be calculated using Ampere's Law or the Biot-Savart Law, and depends on the current \( I_1 \) in coil 1.
4. Step 4: Determine \( \frac{d\Phi_B}{dt} \). The rate of change of magnetic flux \( \frac{d\Phi_B}{dt} \) depends on the rate of change of the current \( \frac{dI_1}{dt} \) in coil 1, as shown in the graph. Use the graph to find \( \frac{dI_1}{dt} \) at \( t = 0.05 \, \text{s} \) and \( t = 0.25 \, \text{s} \). Substitute this into the expression for \( \frac{d\Phi_B}{dt} \).
5. Step 5: Relate the induced EMF to the current in coil 2. The induced EMF drives a current \( I_2 \) in coil 2, which can be calculated using Ohm's Law: \( I_2 = \frac{\text{EMF}}{R} \), where \( R \) is the resistance of coil 2. The resistance \( R \) can be calculated using the wire's diameter and resistivity. Substitute the values to find \( I_2 \) at \( t = 0.05 \, \text{s} \) and \( t = 0.25 \, \text{s} \).

Key Concepts

Electromagnetic Induction

Electromagnetic induction is the process by which a changing magnetic field within a coil of wire induces an electromotive force (EMF) in that coil. This principle, described by Faraday's Law, states that the induced EMF is proportional to the rate of change of the magnetic flux through the coil. In this scenario, the changing current in coil 1 creates a varying magnetic field that affects coil 2, leading to an induced current.

Mutual Inductance

Mutual inductance is a measure of the ability of one coil to induce an EMF in another nearby coil due to a changing current. It depends on the physical arrangement of the coils and the magnetic permeability of the medium between them. The mutual inductance between coil 1 and coil 2 will determine how much current is induced in coil 2 when the current in coil 1 changes over time.

Lenz's Law

Lenz's Law states that the direction of the induced current in a coil will be such that it opposes the change in magnetic flux that produced it. This means that if the current in coil 1 increases, the induced current in coil 2 will flow in a direction that creates a magnetic field opposing the increase. Understanding Lenz's Law is crucial for determining the direction of the induced current in coil 2 at the specified times.