Ch 30: Electromagnetic Induction

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 30: Electromagnetic Induction

Problem 58a

Chapter 30, Problem 58a

INT You've decided to make the magnetic projectile launcher shown in FIGURE P30.58 for your science project. An aluminum bar slides along metal rails through a magnetic field B. The switch closes at t = 0 s, while the bar is at rest, and a battery of emf ε_bat starts a current flowing around the loop. The battery has internal resistance r. The resistances of the rails, which are separated by distance l, and the bar are effectively zero. Show that the bar reaches a terminal speed v_term, and find an expression for v_term.

Verified step by step guidance

Step 1: Begin by analyzing the forces acting on the aluminum bar. The bar moves through a magnetic field \( \mathbf{B} \), which induces a magnetic force due to the current flowing through the loop. This force is given by \( F_{\text{mag}} = I l B \), where \( I \) is the current, \( l \) is the length of the bar, and \( B \) is the magnetic field strength.

Step 2: The current \( I \) in the circuit is determined by Ohm's law: \( I = \frac{\varepsilon_{\text{bat}}}{r} \), where \( \varepsilon_{\text{bat}} \) is the emf of the battery and \( r \) is the internal resistance of the battery. Substitute this expression for \( I \) into the magnetic force equation.

Step 3: As the bar accelerates, it induces a motional emf \( \varepsilon_{\text{motional}} \) due to its motion through the magnetic field. This emf opposes the battery emf and is given by \( \varepsilon_{\text{motional}} = v l B \), where \( v \) is the velocity of the bar. The net emf in the circuit becomes \( \varepsilon_{\text{net}} = \varepsilon_{\text{bat}} - \varepsilon_{\text{motional}} \).

Step 4: The current in the circuit now becomes \( I = \frac{\varepsilon_{\text{net}}}{r} = \frac{\varepsilon_{\text{bat}} - v l B}{r} \). Substitute this expression for \( I \) into the magnetic force equation \( F_{\text{mag}} = I l B \). The magnetic force now depends on the velocity \( v \) of the bar.

Step 5: At terminal velocity \( v_{\text{term}} \), the magnetic force \( F_{\text{mag}} \) balances the resistive force (or any opposing forces, such as friction). Solve for \( v_{\text{term}} \) by setting \( F_{\text{mag}} = 0 \) when the net emf becomes zero. The final expression for \( v_{\text{term}} \) is \( v_{\text{term}} = \frac{\varepsilon_{\text{bat}}}{l B} \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Electromagnetic Induction

Electromagnetic induction is the process by which a changing magnetic field within a closed loop induces an electromotive force (emf) in the loop. This principle, described by Faraday's law, is fundamental in understanding how the aluminum bar in the launcher experiences a force when it moves through the magnetic field, leading to its acceleration.

Terminal Velocity

Terminal velocity refers to the constant speed that an object eventually reaches when the forces acting upon it are balanced. In the context of the magnetic projectile launcher, the bar will reach a terminal speed when the magnetic force opposing its motion equals the net force due to the battery's emf and internal resistance, resulting in no further acceleration.

Ohm's Law

Ohm's Law states that the current flowing through a conductor between two points is directly proportional to the voltage across the two points and inversely proportional to the resistance. In this scenario, it helps to determine the current flowing in the circuit created by the battery, the internal resistance, and the induced emf, which are crucial for calculating the forces acting on the bar.

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Related Practice

Textbook Question

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. If the wire is pulled for 10 s, what is the temperature increase of the carbon? The specific heat of carbon is 710 J/kg K.

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Textbook Question

INT You've decided to make the magnetic projectile launcher shown in FIGURE P30.58 for your science project. An aluminum bar slides along metal rails through a magnetic field B. The switch closes at t = 0 s, while the bar is at rest, and a battery of emf ε_bat starts a current flowing around the loop. The battery has internal resistance r. The resistances of the rails, which are separated by distance l, and the bar are effectively zero. Evaluate v_term for ε_bat= 1.0 V, r = 0.10 Ω, l = 6.0 cm, and B = 0.50 T.

Textbook Question

CALC Your camping buddy has an idea for a light to go inside your tent. He happens to have a powerful (and heavy!) horseshoe magnet that he bought at a surplus store. This magnet creates a 0.20 T field between two pole tips 10 cm apart. His idea is to build the hand-cranked generator shown in FIGURE P30.57. He thinks you can make enough current to fully light a 1.0 Ω lightbulb rated at 4.0 W. That's not super bright, but it should be plenty of light for routine activities in the tent. Find an expression for the induced current as a function of time if you turn the crank at frequency f. Assume that the semicircle is at its highest point at t = 0 s.

Textbook Question

INT FIGURE P30.59 shows a U-shaped conducting rail that is oriented vertically in a horizontal magnetic field. The rail has no electric resistance and does not move. A slide wire with mass m and resistance R can slide up and down without friction while maintaining electrical contact with the rail. The slide wire is released from rest. Determine the value of v_term if l = 20 cm,m = 10 g, R = 0.10 Ω, and B = 0.50 T.

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Textbook Question

CALC Your camping buddy has an idea for a light to go inside your tent. He happens to have a powerful (and heavy!) horseshoe magnet that he bought at a surplus store. This magnet creates a 0.20 T field between two pole tips 10 cm apart. His idea is to build the hand-cranked generator shown in FIGURE P30.57. He thinks you can make enough current to fully light a 1.0 Ω lightbulb rated at 4.0 W. That's not super bright, but it should be plenty of light for routine activities in the tent. With what frequency will you have to turn the crank for the maximum current to fully light the bulb? Is this feasible?

Textbook Question

One way to measure the strength of a magnetic field is with a flip coil. Suppose a 200-turn, 4.0-cm-diameter coil with a resistance of 2.0 Ω is connected to a ballistic galvanometer, a device that measures the total charge passing through. The coil is held perpendicular to the field, then quickly flipped 180° so that the opposite side is facing the magnetic field. Afterward, the galvanometer reads 7.5 μC. What is the field strength? Hint: Use I = dq/dt to relate the net change of flux to the amount of charge that flows through the galvanometer.

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