A square loop is exiting a constant and uniform magnetic field B with a constant velocity v perp

A square loop is exiting a constant and uniform magnetic...

A square loop is exiting a constant and uniform magnetic field $\boldsymbol{B}$ with a constant velocity $v$ perpendicular to the magnetic field. Find the emf induced in the loop when the left end of the loop is at a distance $x$ from the right boundary of the magnetic field. Find the force required to maintain the velocity of the loop. The loop has resistance $R$ and negligible self-inductance. (GRAPH CAN'T COPY)

Key Concepts

Faraday's Law of Induction

Faraday's Law states that a changing magnetic flux through a closed loop induces an electromotive force (emf) in that loop. This is the core principle that explains how motion relative to a magnetic field can lead to the generation of electrical energy in conductive circuits.

Magnetic Flux

Magnetic flux is the measure of the total magnetic field passing through a given area, often represented as the product of the magnetic field strength and the area perpendicular to the field. In problems involving moving loops in magnetic fields, changes in this flux due to movement or altering areas lead to the induction of emf.

Motional EMF

Motional EMF arises when a conductor moves through a magnetic field, causing the free charges within it to experience a Lorentz force. This process separates charges and produces a potential difference along the conductor, which is a manifestation of electromagnetic induction and a direct application of Faraday's Law.

Lenz's Law

Lenz's Law determines the direction of the induced current resulting from a change in magnetic flux. It states that the induced current will flow in such a way that its own magnetic field opposes the change that created it, thereby conserving energy and enforcing the principle of action and reaction in electromagnetic systems.

Ohm's Law

Ohm's Law relates the current flowing through a circuit to the voltage across it and its resistance, with the current being equal to the voltage divided by the resistance. In the context of induced emf in a resistive loop, Ohm's Law is used to calculate the current produced by the emf generated when the loop moves relative to the magnetic field.

Force on a Moving Conductor in a Magnetic Field

When a current-carrying conductor moves in a magnetic field, it experiences a magnetic force that tends to oppose the motion (as explained by Lenz's Law). This force must be counteracted by an external force to maintain constant velocity. The analysis of this force involves understanding the interaction between the magnetic field and the induced current, which is crucial in determining the power requirements for the motion.

Transcript

00:02 Hi, in the given problem the magnetic field, the time varying magnetic field, which is given as b, as a function of time, is equal to a plus b multiplied by t, where a and b are the constants.

00:20 And this magnetic field is linked with a circular coil having a radius r.

00:30 So, in the whole, first part of the problem we have to find the magnetic flux linked with the coil so to find this magnetic flux we use the simple expression for flux by b as the product of magnetic field with the area of the coil as we have to find the magnetic flux linked with the coil at a time zero so here we will write it a plus b multiplied by t into area which is pi r square so for the time to be zero it will come out to be a plus b into zero into pi r square so finally the expression for this magnetic flux linked with the coil comes out to be a pi r square and it becomes the answer for the first part of the problem in the second part of the problem we have to find an expression for the emf induced in this coil for which we will use faraday's laws of electromagnetic induction which says emf induced is given as the time negative of time rate of change of magnetic flux linked through the coil so this is minus d 5 b by d t here it becomes the time derivative of the flux the magnetic flux which is given as a plus b t into pi r square for area taking this pi r square as a constant out it becomes the time derivative of a plus b t then we differentiate this bracket in the first term there is only a constant so its differentiation will come out to be zero in the second term it is the differentiation of b into t so it will just come out to be b only so the expression for mf induced will become minus pi into r square into b and it becomes the answer for the second part of the problem now in the third part of the problem we have to find the expression for current passing through this coil so using home's law the expression for current will come out to be emf divided by resistance in the resistance of the coil is given as r so here i comes out to be minus by r is square b divided by the resistance r...

Please give Ace some feedback