INT A 20-cm-long, zero-resistance slide wire moves outward, on zero-resistance rails, at a steady speed of 10 m/s in a 0.10 T magnetic field. (See Figure 30.26.) On the opposite side, a 1.0 Ω carbon resistor completes the circuit by connecting the two rails. The mass of the resistor is 50 mg. What is the induced current in the circuit?

Verified step by step guidance

1. Step 1: Understand the concept of electromagnetic induction. When a conductor moves through a magnetic field, an electromotive force (EMF) is induced in the conductor due to Faraday's law of induction. The EMF can be calculated using the formula: $\epsilon =Blv$, where $B$ is the magnetic field strength, $l$ is the length of the wire, and $v$ is the velocity of the wire.
2. Step 2: Substitute the given values into the formula for EMF. The magnetic field strength $B$ is 0.10 T, the length of the wire $l$ is 20 cm (convert to meters: $0.20$ m), and the velocity $v$ is 10 m/s. Calculate the EMF using $\epsilon =Blv$.
3. Step 3: Use Ohm's law to find the induced current in the circuit. Ohm's law states that $I=\epsilon /R$, where $I$ is the current, $\epsilon$ is the EMF, and $R$ is the resistance. The resistance of the carbon resistor is given as 1.0 Ω. Substitute the calculated EMF and the resistance into the formula to find the current.
4. Step 4: Verify the direction of the induced current using Lenz's law. Lenz's law states that the induced current will flow in a direction that opposes the change in magnetic flux. Analyze the setup of the circuit and the motion of the wire to determine the direction of the current.
5. Step 5: Consider the physical implications of the induced current. The resistor will dissipate energy as heat due to the current flowing through it. This energy dissipation can be calculated using the formula for power: $P=I{R}^2$, where $P$ is the power, $I$ is the current, and $R$ is the resistance.

Key Concepts

Faraday's Law of Electromagnetic Induction

Faraday's Law states that a change in magnetic flux through a circuit induces an electromotive force (EMF) in that circuit. The induced EMF is proportional to the rate of change of the magnetic flux. In this scenario, the movement of the slide wire through the magnetic field creates a change in flux, leading to the generation of EMF and subsequently inducing current in the circuit.

Ohm's Law

Ohm's Law relates the voltage (V), current (I), and resistance (R) in an electrical circuit, expressed as V = IR. This fundamental principle allows us to calculate the current flowing through a circuit when the voltage and resistance are known. In this problem, once the induced EMF is determined, Ohm's Law can be applied to find the induced current through the 1.0 Ω resistor.

Magnetic Force on a Current-Carrying Conductor

When a conductor moves through a magnetic field, it experiences a magnetic force that can affect the current flowing through it. The direction of this force is given by the right-hand rule, which helps determine the orientation of the induced current. Understanding this concept is crucial for analyzing how the motion of the slide wire in the magnetic field influences the overall circuit behavior.