FIGURE P30.47 shows a 1.0-cm-diameter loop with R = 0.50 Ω inside a 2.0-cm-diameter solenoid. The solenoid is 8.0 cm long, has 120 turns, and carries the current shown in the graph. A positive current is cw when seen from the left. Determine the current in the loop at t = 0.010 s.

Verified step by step guidance

1. Step 1: Understand the setup. The solenoid has a diameter of 2.0 cm, a length of 8.0 cm, and 120 turns. The loop inside the solenoid has a diameter of 1.0 cm and a resistance of 0.50 Ω. The graph shows the current in the solenoid, which varies linearly with time. At t = 0.010 s, the solenoid current is 0 A, and its rate of change can be determined from the slope of the graph.
2. Step 2: Calculate the magnetic field inside the solenoid. The magnetic field inside a solenoid is given by the formula: $B=\frac{\mu ₀nI}{L}$, where $\mu ₀$ is the permeability of free space, $n$ is the number of turns per unit length, and $I$ is the current in the solenoid. First, calculate $n$ as $n=\frac{\mathrm{120}}{\mathrm{0.08}}$ turns per meter.
3. Step 3: Determine the rate of change of the magnetic field. The graph shows that the solenoid current changes linearly with time. The slope of the graph gives the rate of change of current, $\frac{\mathrm{dI}}{\mathrm{dt}}$. Use this slope to calculate the rate of change of the magnetic field, $\frac{\mathrm{dB}}{\mathrm{dt}}$, using the formula: $\frac{\mathrm{dB}}{\mathrm{dt}}=\mu ₀n\frac{\mathrm{dI}}{\mathrm{dt}}$.
4. Step 4: Calculate the induced emf in the loop. The induced emf is given by Faraday's law: $\epsilon =-\frac{\mathrm{d\Phi }}{\mathrm{dt}}$, where $\Phi$ is the magnetic flux. The flux through the loop is $\Phi =BA$, where $A$ is the area of the loop. Use the formula for the area of a circle, $A=\pi r²$, to calculate the area of the loop, and substitute $\frac{\mathrm{dB}}{\mathrm{dt}}$ to find $\epsilon$.
5. Step 5: Determine the current in the loop. The induced current in the loop is given by Ohm's law: $I=\frac{\epsilon }{R}$, where $R$ is the resistance of the loop. Substitute the value of $\epsilon$ and $R$ to calculate the current in the loop at $t=0.010$ s.

Key Concepts

Electromagnetic Induction

Electromagnetic induction is the process by which a changing magnetic field within a closed loop induces an electromotive force (EMF) in that loop. This phenomenon is described by Faraday's Law, which states that the induced EMF is proportional to the rate of change of magnetic flux through the loop. In this scenario, the current in the solenoid creates a magnetic field that changes over time, inducing a current in the loop.

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 the loop, once the induced EMF is calculated, Ohm's Law can be used to determine the current flowing through the loop by dividing the induced EMF by the resistance of the loop. This relationship is crucial for solving the problem as it connects the induced voltage to the resulting current.

Magnetic Flux

Magnetic flux is a measure of the quantity of magnetism, taking into account the strength and the extent of a magnetic field. It is defined as the product of the magnetic field (B) and the area (A) through which the field lines pass, and is given by the equation Φ = B·A·cos(θ), where θ is the angle between the magnetic field lines and the normal to the surface. Understanding magnetic flux is essential for determining how the changing magnetic field from the solenoid affects the loop.