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.

Verified step by step guidance

Step 1: Understand the problem setup. The slide wire is moving vertically in a horizontal magnetic field, generating an electromotive force (EMF) due to its motion. This EMF induces a current in the circuit formed by the rail and the wire. The magnetic field exerts a force on the current-carrying wire, opposing its motion. At terminal velocity, the magnetic force balances the gravitational force acting on the wire.

Step 2: Write the expression for the EMF induced in the wire. The EMF (ε) is given by Faraday's law: ε = B * l * v, where B is the magnetic field strength, l is the length of the wire, and v is the velocity of the wire.

Step 3: Determine the current in the circuit. Using Ohm's law, the current (I) in the circuit is given by I = ε / R, where R is the resistance of the wire. Substituting ε = B * l * v, we get I = (B * l * v) / R.

Step 4: Calculate the magnetic force acting on the wire. The magnetic force (F_mag) is given by F_mag = B * l * I. Substituting I = (B * l * v) / R, we get F_mag = (B^2 * l^2 * v) / R.

Step 5: Set up the force balance equation at terminal velocity. At terminal velocity, the magnetic force balances the gravitational force: F_mag = F_gravity. The gravitational force is given by F_gravity = m * g, where m is the mass of the wire and g is the acceleration due to gravity. Equating F_mag and F_gravity, we get (B^2 * l^2 * v_term) / R = m * g. Solve this equation for v_term 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 wire. This principle, described by Faraday's law, states that the induced EMF is proportional to the rate of change of the magnetic flux through the loop. In this scenario, as the slide wire moves through the magnetic field, it experiences a change in magnetic flux, leading to the generation of an induced current.

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 F is the force, q is the charge, v is the velocity of the particle, and B is the magnetic field. In this case, the current induced in the slide wire creates a magnetic force that acts perpendicular to both the direction of the current and the magnetic field, influencing the motion of the wire.

Terminal Velocity

Terminal velocity is the constant speed that a freely falling object eventually reaches when the resistance of the medium prevents further acceleration. In this context, the slide wire will reach a terminal velocity when the gravitational force acting on it is balanced by the magnetic force due to the induced current. This balance of forces allows us to calculate the terminal velocity using the mass of the wire, the resistance, and the magnetic field strength.

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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

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

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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

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Textbook Question

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