A circular loop of wire with radius r=0.0480 m and resistance R=0.160 Ωis in a region of spatial

A circular loop of wire with radius r=0.0480 m and...

A circular loop of wire with radius $r=0.0480 \mathrm{m}$ and resistance $R=0.160 \Omega$ is in a region of spatially uniform magnetic field, as shown in Fig. E29. $20 .$ The magnetic field is directed out of the plane of the figure. The magnetic field has an initial value of 8.00 $\mathrm{T}$ and is decreasing at a rate of $d B / d t=$ $-0.680 \mathrm{T} / \mathrm{s}$ (a) Is the induced current in the loop clockwise or counterclockwise? (b) What is the rate at which electrical energy is being dissipated by the resistance of the loop?

A circular loop of wire with radius $r=0.0480 \mathrm{m}$ and resistance $R=0.160 \Omega$ is in a region of spatially uniform magnetic field, as shown in Fig. E29. $20 .$ The magnetic field is directed out
of the plane of the figure. The magnetic field has an initial value of 8.00 $\mathrm{T}$ and is decreasing at a rate of $d B / d t=$ $-0.680 \mathrm{T} / \mathrm{s}$ (a) Is the induced current in the loop clockwise or counterclockwise? (b) What is the rate at which electrical energy is being dissipated by
the resistance of the loop?

Key Concepts

Faraday's Law of Electromagnetic Induction

Faraday's Law states that a changing magnetic flux through a closed loop induces an electromotive force (EMF) in the loop. This fundamental law underpins the operation of many electrical devices and explains how electrical energy can be generated from a changing magnetic environment.

Magnetic Flux

Magnetic flux is the measure of the amount of magnetic field passing through a given area. It is calculated as the product of the magnetic field strength and the perpendicular area it penetrates, and its change over time is directly linked to the induced EMF in a circuit as described by Faraday's Law.

Lenz's Law

Lenz's Law provides the directional aspect of the induced electromotive force and current, stating that the induced current will flow in such a direction as to oppose the change in magnetic flux that produced it. This principle is essential for understanding the behavior of induced currents in circuits.

Ohm's Law

Ohm's Law relates the voltage, current, and resistance in an electrical circuit. It is used to determine the current that results from an induced EMF in a circuit with a given resistance, forming the basis for analyzing electrical circuits involving induced currents.

Resistive Power Dissipation

Resistive power dissipation refers to the conversion of electrical energy into thermal energy in a resistor. This concept is quantified by the relation P = I²R, which describes how the power loss in an electrical circuit depends on the current flowing through the circuit and the resistance it encounters.

Transcript

00:02 Hi, in the given problem this is the circular loop of a conducting wire and there is a magnetic field coming out of the plane of paper and this is gling through this coil also like this the radius of this coil has been given as 0 .0 480 meter the resistance of the coil is 0 .160 meter the resistance of the coil is 0 .1 660 om and the magnetic field is coming outward and this magnetic field is reducing with the time at the given rate as minus 0 .680 tesla per second and the initial magnetic field linked with the coil is b0 is equal to 8 .00 tesla in the first part of the problem we have to find the direction of the of current induced in this coil as a result of change of magnetic field as the magnetic field is outward and it is decreasing so the reason behind the emf induced will be outward decreasing magnetic flux so as per lens's law the induced magnetic field should be in a direction which may oppose this outward decreasing magnetic flux so the induced magnetic flux will be outward but increasing as the magnetic flux induced magnetic flux is outward increasing so this face the upper face of the coil should start behaving like north pole or we can say the direction of current induced in this should be counterclockwise here this is the direction of the current induced which is counterclockwise and this is the answer for the first part of the problem now in the second part of the problem we have to find the rate at which electrical energy is being dissipated by the resistance of the loop so for this purpose first of all we will find the expression the value the magnitude of emf induced and that is given by faraday's laws of electromagnetic induction as emf induced is minus d -5 by d -t or we can say this is the differentiation of b times a where a is the area and b is the magnetic field with respect to time or it will become as the area is constant so this is minus a d b by d t so we put these values here this is pi r square and we know the radius is 0 .048 to the whole square and for db by d t we know this is minus 0 .680 so finally this emf induced comes out to be 4 .94 into 10 dash to the power minus 3 volt this is the magnitude of the emf induced in this coil now finally we have to find the rate at which the electrical energy is being dissipated for which we will use the formula for the power as emf squared divided by the resistance and it comes out to be 4 .94 multiplied by 10 dash to power minus 3 to the whole square divided by resistance which was 0 .1 60 and finally this power comes out to be 1 .52 multiplied by 10 dashed bar minus 4 watt...

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