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.

Verified step by step guidance

Step 1: Understand the setup of the problem. The aluminum bar slides along metal rails in a magnetic field \( \mathbf{B} \). The battery with emf \( \varepsilon_{\text{bat}} \) and internal resistance \( r \) drives a current through the loop formed by the rails and the bar. The magnetic field exerts a force on the bar due to the current flowing through it, causing it to accelerate until it reaches a terminal velocity \( v_{\text{term}} \).

Step 2: Recall the formula for the magnetic force acting on the bar. The magnetic force is given by \( F_{\text{mag}} = I \cdot l \cdot B \), where \( I \) is the current, \( l \) is the length of the bar, and \( B \) is the magnetic field strength. This force opposes the motion of the bar and balances the resistive force at terminal velocity.

Step 3: Determine the current \( I \) in the circuit. The current is given by Ohm's law: \( I = \frac{\varepsilon_{\text{bat}}}{r} \), where \( \varepsilon_{\text{bat}} \) is the emf of the battery and \( r \) is its internal resistance.

Step 4: At terminal velocity, the magnetic force \( F_{\text{mag}} \) balances the resistive force due to the motion of the bar. The resistive force is caused by the induced emf \( \varepsilon_{\text{ind}} \) in the bar, which opposes the battery emf. The induced emf is given by \( \varepsilon_{\text{ind}} = v_{\text{term}} \cdot l \cdot B \), where \( v_{\text{term}} \) is the terminal velocity.

Step 5: Set up the balance condition at terminal velocity. The induced emf \( \varepsilon_{\text{ind}} \) equals the battery emf \( \varepsilon_{\text{bat}} \). Substitute \( \varepsilon_{\text{ind}} = v_{\text{term}} \cdot l \cdot B \) and solve for \( v_{\text{term}} \): \( v_{\text{term}} = \frac{\varepsilon_{\text{bat}}}{l \cdot B} \). Plug in the given values for \( \varepsilon_{\text{bat}} \), \( l \), and \( B \) to find the terminal velocity.

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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 motion of the aluminum bar through the magnetic field generates an electric current, which is crucial for the operation of the magnetic projectile launcher.

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, the internal resistance of the battery and the effective resistance of the circuit will determine the current flowing through the loop, which is essential for calculating the terminal velocity of the bar.

Lorentz Force

The Lorentz force is the force experienced by a charged particle moving through a magnetic field. It is given by the equation F = q(v × B), where q is the charge, v is the velocity of the particle, and B is the magnetic field. In the context of the magnetic projectile launcher, this force acts on the current-carrying bar, propelling it along the rails and allowing us to evaluate its terminal velocity.

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Textbook Question

BIO One possible concern with MRI (see Exercise 28) is turning the magnetic field on or off too quickly. Bodily fluids are conductors, and a changing magnetic field could cause electric currents to flow through the patient. Suppose a typical patient has a maximum cross-section area of 0.060 m². What is the smallest time interval in which a 5.0 T magnetic field can be turned on or off if the induced emf around the patient's body must be kept to less than 0.10 V?

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

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.

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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