A metal rod of mass m slides without friction along two...

A metal rod of mass $m$ slides without friction along two parallel horizontal rails, separated by a distance $\ell$ and connected by a resistor $R$, as shown in Figure $\mathrm{P} 23.15$. A uniform vertical magnetic field of magnitude $B$ is applied perpendicular to the plane of the paper. The applied force shown in the figure acts only for a moment, to give the rod a speed $v .$ In terms of $m, \ell, R, B,$ and $v,$ find the distance the rod will then slide as it coasts to a stop.

Key Concepts

Motional Electromotive Force (EMF)

This concept involves the generation of a voltage across a conductor moving through a magnetic field. In a system where a conductive rod moves in the presence of a magnetic field, the Lorentz force acting on the charge carriers produces an electromotive force proportional to the product of the magnetic field strength, the velocity of the rod, and the effective length of the conductor within the magnetic field. This induced voltage is fundamental to understanding how mechanical motion can be converted into electrical energy in such setups.

Lenz's Law

Lenz's Law states that the direction of induced current in a closed circuit due to a changing magnetic flux will oppose the change producing it. In the context of a sliding rod moving through a magnetic field, the induced current creates its own magnetic field which opposes the motion of the rod. This opposition is crucial as it results in a retarding force that slows down the rod and eventually stops it, illustrating the principle of electromagnetic braking.

Electromagnetic Damping

Electromagnetic damping arises from the interplay between induced currents and magnetic fields, leading to a force that opposes the motion of a conductor. As the rod moves, the induced motional EMF drives a current through the circuit, and this current interacts with the external magnetic field to produce a magnetic force opposing the rod's motion. This conversion of kinetic energy into electrical energy, which is then dissipated as heat in the resistor, embodies the principle of damping in electromechanical systems.

Energy Conservation in Electromechanical Systems

In these systems, the mechanical energy of the moving body is gradually converted into electrical energy through electromagnetic induction. The kinetic energy imparted to the rod is countered by the electrical energy dissipated in the resistor due to the induced currents, ensuring that the total energy remains conserved. This concept is essential for analyzing the motion of the rod since it links the initial kinetic energy with the work done against the electromagnetic damping force, ultimately dictating the distance the rod travels before coming to rest.

Transcript

00:01 High in the given problem there are two rails metallic rails over which a metallic rod is sliding it is free to slide over these two rails in the given direction and there is a resistance are joining these two rails there is a magnetic field in the region perpendicular to the plane of paper and directed into it like this.

00:46 Initially a force is applied on this raw metallic rod which gives it a speed v so we can say initially means at time zero the initial speed in the bar in this rod metallic rod is v and we have to find the distance traveled by this rod when the rod will come to rest means vf means the final speed achieved by the raw should be zero and we have to find the total distance traveled by the rod in order to achieve a zero speed this distance suppose this is missing here now, the emf induced in the rod as a result of its motion in the magnetic field will be given by b into v into l, where l is the length of this rod and v is its instantaneous velocity.

02:08 So this is the instantaneous value of emf induced across this rod.

02:14 So the current passing through the circuit will become using omslaw, i equals, to enf induced divided by resistance of the circuit or we can say this is b vl divided by r so finally the force exerted on this rod will be given by the magnetic force will become b i into l or as this force is opposite to the direction of motion so we can say this is negative of b il the force is negative in nature.

02:55 So we can say this is negative of b multiplied by b v l divided by r times of l.

03:03 So finally this force becomes minus b square v l square by r.

03:11 And using newton's second law of motion this force can also be given as mass of the rod into acceleration.

03:20 And and using calculus we know acceleration is given as the derivative of instantaneous velocity...

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