Big Loop, Little Loop A small circular loop of area 2.00 cm^2 is placed in the plane of, and co

Big Loop, Little Loop A small circular loop of area 2.00...

Big Loop, Little Loop A small circular loop of area $2.00 \mathrm{~cm}^{2}$ is placed in the plane of, and concentric with, a large circular loop of radius $1.00 \mathrm{~m}$. The current in the large loop is changed uniformly from 200 A to $-200 \mathrm{~A}$ (a change in direction) in a time of $1.00 \mathrm{~s}$, beginning at $t_{1}=0 .$ (a) What is the magnitude of the magnetic field at the center of the small circular loop due to the current in the large loop at $t_{1}=0 \mathrm{~s}, t_{2}=0.500 \mathrm{~s}$, and $t_{3}=1.00 \mathrm{~s}$ ? (b) What is the magnitude of the emf induced in the small loop at $t_{2}=0.500 \mathrm{~s}$ ? (Since the inner loop is small, assume the ficld $\vec{B}$ due to the outer loop is uniform over the area of the smaller loop.)

Key Concepts

Magnetic Field from a Current Loop

This concept involves the determination of the magnetic field at the center of a circular loop carrying an electric current. Using the Biot–Savart law, the magnetic field is given by B = ??I/(2R), where ?? is the permeability of free space, I is the current through the loop, and R is its radius. This relation is central to calculating the magnetic field produced by a loop and understanding how it varies with current and geometry.

Magnetic Flux

Magnetic flux is defined as the product of the magnetic field and the area through which the field lines pass, considering the angle between the field and the area normal. When the magnetic field is uniform and perpendicular to the surface, the flux simplifies to ? = BA. This concept is key to understanding how variations in the magnetic field or the area (or both) lead to changes in flux, which in turn can induce an electromotive force in a loop.

Faraday's Law of Electromagnetic Induction

Faraday’s law states that a changing magnetic flux through a circuit induces an electromotive force (emf) in that circuit, with the magnitude of the induced emf given by the rate of change of the flux, |?| = |d?/dt|. This law is fundamental for analyzing situations where currents change with time and consequently induce an emf in nearby circuits or loops.

Uniform Magnetic Field Approximation

In many electromagnetic problems, especially when dealing with small loops in the vicinity of larger current-carrying loops, it is often assumed that the magnetic field is uniform over the small area of interest. This approximation simplifies calculations of magnetic flux and induced emf by allowing the use of a constant magnetic field value across the entire area of the loop.

Transcript

00:03 In the given problem the current passing through the bigger loop is changing from 200 ampere in one direction to minus 200 amp and obviously that will be in other direction in opposite direction and that is happening in a time t is equal to 1 .00 second.

00:56 So it is obvious if the time rate of change of current is uniform then in first half means from 0 second to 0 .5 00 second the current should have dropped from 200 ampere to 0 ampere and then in other half means from 0 .50 seconds to 1 .00 second the current must have increased again but in opposite direction minus 200 ampere.

02:48 Area of the smaller loop that is given as 2 .00 centimeter square it will be used to find the flux magnetic flux linked through the smaller loop.

03:03 Loop and radius of the bigger loop is given to us r is equal to 1 .00 meter and this radius will be used to find the magnetic field at the center of the bigger loop.

03:25 So in the first part of the problem using the expression for magnetic field at the center of a current carrying coil the bigger loop means it is given as mu -not i by 2r.

03:40 So at a time, t is equal to 0 second, the magnetic field will come out to be mu not i by 2 r.

03:57 For mu not this is 4 pi into 10 dash power minus 7.

04:02 Tesla meter per ampere current was exactly 200 ampere divided by 2 into radius which was 1 .00 meters.

04:10 So magnetic field at the center of the bigger loop at a time t equals to zero second comes out to be b 4 is equal to 1 .3 into 10 dash per minus 4 which is one of the answer for the first part of this given problem then again in the same part we have to find magnetic field at the center of this loop bigger loop but at a time t is equal to 0 .50 second and at that time we have found already that current will be zero so the magnetic field we at the center of the coil bigger coil at a time 0 .5 second will come out to be 0 .2 second answer for the first part of this problem and finally in the same first part of the problem at a time p is equal to 1 .00 second the magnetic field b1 will be given by as the magnitude of the current in it is again 200 ampere so this is again 4 pi into 10 r .m .m .m .m .m...

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