A horizontal wire is free to slide on the vertical rails of...

A horizontal wire is free to slide on the vertical rails of a conducting frame, as shown in Figure $\mathrm{P} 31.68$. The wire has mass $m$ and length $\ell$, and the resistance of the circuit is $R$. If a uniform magnetic field is directed perpendicular to the frame, what is the terminal speed of the wire as it falls under the force of gravity?

Key Concepts

Ohm's Law

Ohm's law relates the voltage, current, and resistance in an electrical circuit by stating that the current flowing through a conductor is directly proportional to the voltage across it and inversely proportional to its resistance. This law is fundamental in calculating the induced current when an EMF is generated in a circuit with a defined resistance.

Lorentz Force

The Lorentz force is the force experienced by a charged particle or a current-carrying conductor in the presence of a magnetic field. It acts perpendicular to both the direction of the current and the magnetic field, and in systems with induced currents, it can provide an opposing force to the motion of the conductor.

Motional EMF

Motional electromotive force (EMF) is generated when a conductor moves through a magnetic field, causing a separation of charges along its length due to the Lorentz force. This induced voltage is proportional to the magnetic field strength, the length of the conductor within the field, and its velocity relative to the field.

Terminal Speed

Terminal speed is the constant velocity attained by an object when the net force acting on it becomes zero due to a balance between the driving forces (such as gravity) and resisting forces (such as drag or electromagnetic effects). In problems involving falling bodies, it represents the state where acceleration ceases because the upward forces exactly offset the gravitational pull.

Lenz's Law

Lenz's law states that the direction of the induced current from a changing magnetic environment is such that it opposes the change that created it. This principle ensures energy conservation by causing the induced magnetic field to oppose the motion or change that produced the EMF, resulting in a damping effect on the movement.

Transcript

00:01 In this question, they have given that the width of a rectangular frame is equal to d.

00:05 So, our width of frame, it is equal to d and the mass of the rectangular frame is equal to m.

00:16 The mass is equal to m and the resistance of the coil is equal to capital r.

00:26 So in the first part, we have to calculate the net acceleration.

00:29 So, how can we calculate the net acceleration? as we know that, we can say that frame attains the speed v, emf developed in the side ab.

00:40 So first of all, emf developed in the side ab, it will be equal to b multiply by dv and v is the velocity here, right? and current will be equal to, if i find the current, so i can easily say my current will be equal to voltage divided by resistance, so it will be b, multiply by, by d v divided by the resistance capital r and now if i say my magnetic force so as we know that magnetic force is equal to i multiplied by l multiplied by the magnetic field capital b so from this my force will be equal to because we have i so it will be equal to b multiplied by the length is d square multiply by velocity divided by resistance so let's say this is our equation one and as we know that as the force in the direction opposite to that of motion of the frame, therefore the net force is given by because the force is in the direction opposite to the motion.

01:40 So what i can say, that the net force, it will be equal to capital f minus magnetic force and capital f minus magnetic force is equal to bd square v divided by r and this is magnetic force.

01:58 When we will calculate this and know that our f net is equal to small m multiply by a from the newton's second low so after calculating this we will get m a it will be equal to f minus b d square v divided by r so when we will calculate this we will get the net acceleration that will be equal to r f minus b d square v divided by m multiplied by the resistance r so this is the answer of the first part of this question...

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