FIGURE EX30.13 shows a 10-cm-diameter loop in three different magnetic fields. The loop's resistance is 0.20 Ω. For each, what are the size and direction of the induced current?

Verified step by step guidance

1. Step 1: Understand Faraday's Law of Induction, which states that the induced electromotive force (EMF) in a loop is proportional to the rate of change of magnetic flux through the loop. The formula is: $\epsilon =-\frac{d\Phi B}{\mathrm{dt}}$, where $\Phi B$ is the magnetic flux.
2. Step 2: Calculate the magnetic flux for each case. Magnetic flux is given by the formula: $\Phi B=BA\cos \theta$, where $B$ is the magnetic field strength, $A$ is the area of the loop, and $\theta$ is the angle between the magnetic field and the normal to the loop.
3. Step 3: Determine the rate of change of magnetic flux ($\frac{d}{\Phi }\mathrm{dt}$) for each scenario. If the magnetic field is changing, calculate the change in $B$ over time. If the field is constant, the induced EMF will be zero.
4. Step 4: Use Ohm's Law to calculate the induced current. The formula is: $I=\frac{\epsilon }{R}$, where $\epsilon$ is the induced EMF and $R$ is the resistance of the loop (given as 0.20 Ω).
5. Step 5: Determine 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 direction of the changing magnetic field in each case to determine the current's direction.

Key Concepts

Faraday's Law of Electromagnetic Induction

Faraday's Law states that a change in magnetic flux through a loop induces an electromotive force (EMF) in the loop. The induced EMF is proportional to the rate of change of the magnetic flux, which can occur due to a changing magnetic field or the movement of the loop within a magnetic field. This principle is fundamental for understanding how currents are generated in conductive loops.

Lenz's Law

Lenz's Law provides the direction of the induced current resulting from electromagnetic induction. It states that the induced current will flow in a direction that opposes the change in magnetic flux that produced it. This law ensures the conservation of energy and helps predict the behavior of induced currents in various scenarios.

Ohm's Law

Ohm's Law relates the voltage (V), current (I), and resistance (R) in an electrical circuit, expressed as V = IR. In the context of induced currents, once the induced EMF is calculated using Faraday's Law, Ohm's Law can be used to determine the size of the current flowing through the loop by dividing the induced EMF by the loop's resistance.