OA11 Magnetic Fields, Ampere\'s Law, Electromagnetic induction PDF

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11/15/2020

OA11: Magnetic Fields, Ampere's Law, Electromagnetic induction

OA11: Magnetic Fields, Ampere's Law, Electromagnetic induction Due: 11:59pm on Sunday, November 8, 2020 You will receive no credit for items you complete after the assignment is due. Grading Policy

Problem 28.08

Part A The magnetic field at a distance of 2 cm from a current carrying wire is 4 μT. What is the magnetic field at a distance of 4 cm from the wire? ANSWER: 2 μT 10 μT 4 μT 6 μT 8 μT

Correct

Magnetic Field due to Semicircular Wires A loop of wire is in the shape of two concentric semicircles as shown. The inner circle has radius ; the outer circle has radius . A current flows clockwise through the outer wire and counterclockwise through the inner wire.

Part A What is the magnitude, Express

, of the magnetic field at the center of the semicircles?

in terms of any or all of the following: , , , and

.

Hint 1. What physical principle to use https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=8572633

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Use the Biot-Savart law separately on each semicircle and on the wire segments that join the semicircles. You should then add the results. This is possible because the magnetic field is a vector field and hence it is additive. The BiotSavart law is

where is the permeability of free space and point where the magnetic field is measured.

is the distance between the infinitesimal wire element

and the

Hint 2. Compute the field due to the inner semicircle What is the magnitude,

, of the magnetic field due to the inner semicircle (of radius ) at the point P?

Express your answer in terms of , , and

.

Hint 1. Finding the integrand Pulling the constants outside of the integral we have . What is the value of the integrand Express

in terms of

for the semicircle? and .

Hint 1. Determine the direction of the field due to any point on the inner semicircle What is the direction of this field? ANSWER:

into the screen out of the screen

Hint 2. The cross product Observe that

is just

along the tangent and

in this case, since they are perpendicular to each other with

lying

being the radial vector.

Hint 3. The -dependence of Recall that in this case

, independent of which point on the semicircle you are considering.

ANSWER:

=

Hint 2. Evaluate the integral In the last part you found that the integral simplifies to

. What is

over the semicircle with

radius ? Express your answer in terms of

and standard constants.

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Hint 1. How to approach this problem This integral is nothing but the length of the circumference of the semicircle.

ANSWER: =

ANSWER: =

Hint 3. Direction of the field due to the inner semicircle What is the direction of this field? ANSWER:

into the screen out of the screen

Hint 4. Compute the field due to the straight wire segments What is the field due to the two straight wire segments? Express your answer in terms of some or all of the following: , , , and

.

ANSWER: = 0

ANSWER: =

Correct To see whether and makes sense, think of the scaling of different quantities. The size of the current element scales as the radius, whereas the power of in the denominator is 2 (and equals the radius also, in this case). So over all, you would expect the magnetic field to scale as 1/radius. Note that such an argument works only because the field due to each point is in the same direction, so you are doing a much simpler integral.

Part B What is the direction of the magnetic field at the center of the semicircles? https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=8572633

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ANSWER: into the screen out of the screen

Correct

Ampère’s Law Explained Learning Goal: To understand Ampère’s law and its application. Ampère’s law is often written

.

Part A The integral on the left is ANSWER:

the integral throughout the chosen volume. the surface integral over the open surface. the surface integral over the closed surface bounded by the loop. the line integral along the closed loop. the line integral from start to finish.

Correct

Part B What physical property does the symbol

represent?

ANSWER:

The current along the path in the same direction as the magnetic field The current in the path in the opposite direction from the magnetic field The total current passing through the loop in either direction The net current through the loop

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Correct The positive direction of the line integral and the positive direction for the current are related by the right-hand rule: Wrap your right-hand fingers around the closed path, then the direction of your fingers is the positive direction for and the direction of your thumb is the positive direction for the net current. Note also that the angle the current-carrying wire makes with the surface enclosed by the loop doesn't matter. (If the wire is at an angle, the normal component of the current is decreased, but the area of intersection of the wire and the surface is correspondingly increased.)

Part C The circle on the integral means that

must be integrated

ANSWER:

over a circle or a sphere. along any closed path that you choose. along the path of a closed physical conductor. over the surface bounded by the current-carrying wire.

Correct

Part D Which of the following choices of path allow you to use Ampère’s law to analytically find

?

a. The path must pass through the point . b. The path must have enough symmetry so that c. The path must be a circle.

is constant along large parts of it.

ANSWER:

a only a and b a and c b and c

Correct

Part E Ampère’s law can be used to analytically find the magnetic field around a straight current-carrying wire. Is this statement true or false? https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=8572633

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ANSWER: true false

Correct In fact Ampère's law can be used to find the magnetic field inside a cylindrical conductor (i.e., at a radius less than the radius of the wire, ). In this case is just that current inside , not the current inside (which is the total current in the wire).

Part F Ampère’s law can be used to analytically find the magnetic field at the center of a square loop carrying a constant current. Is this statement true or false? ANSWER:

true false

Correct The key point is that to be able to use Ampère's law, the path along which you take the line integral of must have sufficient symmetry to allow you to pull the magnitude of outside the integral. Whether the current distribution has symmetry is incidental.

Part G Ampère’s law can be used to analytically find the magnetic field at the center of a circle formed by a current-carrying conductor. Is this statement true or false? ANSWER: true false

Correct

Part H Ampère’s law can be used to analytically find the magnetic field inside a toroid. (A toroid is a doughnut shape wound uniformly with many turns of wire.) Is this statement true or false? https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=8572633

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ANSWER: true false

Correct Therefore, though Ampère’s law holds quite generally, it is useful in finding the magnetic field only in some cases, when a suitable path through the point of interest exists, typically such that all other points on the path have the same magnetic field through them.

Introduction to Faraday's Law Learning Goal: To understand the terms in Faraday's law for magnetic induction of electric fields, and contrast these fields with those produced by static charges. Faraday's law describes how electric fields and electromotive forces are generated from changing magnetic fields. It relates the line integral of the electric field around a closed loop to the change in the total magnetic field integral across a surface bounded by that loop: , where

is the line integral of the electric field, and the magnetic flux is given by ,

where

is the angle between the magnetic field

and the local normal to the surface bounded by the closed loop.

Direction: The line integral and surface integral reverse their signs if the reference direction of

or

is reversed. The right-hand rule

applies here: If the thumb of your right hand points along

, then the

fingers point along . You are free to take the loop anywhere you choose, although usually it makes sense to choose it to lie along the path of the circuit you are considering.

Part A Consider the direction of the electric field in the figure. Assume that the magnetic field points upward, as shown. Under wha circumstances is the direction of the electric field shown in the figure correct?

Hint 1. How to approach the problem https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=8572633

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There are two approaches: One approach is to realize that the electric field would produce a current in the same direction as the field if there were a wire present. Find the magnetic field due to this (imaginary) current using the right-hand rule. The magnetic field produced by this current must oppose the change in the original magnetic field. You should be able to tell what the change is. The other approach is more mathematical. Choose a direction for

and

. Suppose

is in the direction of the

electric field shown. Then has the same direction as that of the magnetic field shown. The left-hand side of Faraday's equation is positive, and so is the magnetic flux. However, since the left-hand side is positive, the change in flux should be negative. What must have happened to the field?

ANSWER:

always if

increases with time

if

decreases with time

depending on whether your right thumb is pointing up or down

Correct

Part B Now consider the magnetic flux through a surface bounded by the loop. Which of the following statements about this surfac must be true if you want to use Faraday's law to relate the magnetic flux to the line integral of the electric field around the loop? ANSWER: The surface must be the circular disk in the middle of the loop. The surface must be perpendicular to the magnetic field at each point. The surface can be any surface whose edge is the loop. The surface can be any surface whose edge is the loop as long as no magnetic field line passes through it more than once.

Correct You are free to take any surface bounded by the loop as the surface over which to evaluate the integral. The result will always be the same, owing to the continuity of magnetic field lines (they never start or end anywhere, since there are no magnetic charges).

It is important to understand the vast differences between electric fields produced by changing magnetic fields via Faraday's law and the more familiar electric fields produced by charges via Coulomb's law. Here are some short questions that illustrate these differences.

Part C https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=8572633

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When can an electric field be measured at any point from the force on a stationary test charge at that point?

Hint 1. Force on a stationary charge Recall that the total force

on a charge

is .

If the charge is stationary, i.e.,

, then the force reduces to

.

ANSWER:

only if the field is generated by the coulomb field of static charges only if the field is generated by a changing magnetic field no matter how the field is generated

Correct In fact, this operation defines an electric field. Similarly, if the test charge is moving, it will measure magnetic fields.

Part D When can an electric field that does not vary in time arise? ANSWER:

only if the field is generated by a coulomb field of static charges only if the field is generated by a changing magnetic field in either of the above two cases Electric fields never vary in time; otherwise, a charge could gain energy from the field.

Correct

Part E When will the integral

around any closed loop be zero?

ANSWER:

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only if the field is generated by the coulomb field of static charges only if the field is generated by the coulomb field of static charges or a constant current only if the field is generated by a changing magnetic field however the field is generated The loop integral is always zero; otherwise, a charge moving around the loop would gain energy.

Correct The electric field generated by a static charge or a constant current always has zero loop integral. Even when the constant current takes the form of a continuous line of evenly-spaced charges moving with constant velocity, the resulting electric field will have a zero loop integral. An electric field generated by any other configuration of moving charges (moving through the loop) would have a non-zero loop integral.

Here is a simple quantitative problem that uses Faraday's law.

Part F A cylindrical iron rod of infinite length with cross-sectional area is oriented with its axis of symmetry coincident with the z axis of a cylindrical coordinate system as shown in the figure. It has a magnetic field inside that varies according to . Find the theta component of the electric field at distance from the z axis, where is larger than the radius of the rod. Express your answer in terms of , needed constants such as , , and

,

,

, and any

.

Hint 1. Selecting the loop You want to find , but Faraday's law determines only an integral of the field. Therefore, you must pick the loop around which you will integrate in such a way that it involves only one value of and has a constant projection of

along it. Such a loop is a circle with fixed

(at any z, since the rod extends infinitely in the z direction).

Hint 2. Find the magnetic flux Faraday's law states that What is the magnetic flux

. Therefore, to find the electric field you must first find the magnetic flux. at time ?

Express your answer in terms of the magnetic field variables

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and

,

, and .

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Hint 1. The flux integral The flux integral for a constant field is the product of the magnitude of the perpendicular magnetic field and the area over which the integral is evaluated.

ANSWER: =

Correct

Hint 3. Finding the EMF from Faraday's law From the previous part, you found that your equation for

. According to Faraday's law,

to find the EMF

Express your answer in terms of

,

. Use

predicted by Faraday's law. ,

, , and any necessary constants.

ANSWER: =

Correct

Hint 4. Help from symmetry The rod extends along the entire z axis, so the electric field cannot depend on . Since the rod is cylindrically symmetric, the field cannot depend on either. Thus, the component of the electric field is a function only of

.

Hint 5. Find the EMF in terms of By definition, the EMF around a loop is given by component of Express

parallel to in terms of

. The dot product means that you need consider only the

. Find the relationship between

and

.

, quantities given in the introduction to this part, and familiar constants.

Hint 1. The component of the field along the loop Since the loop is a circle with constant along the loop.

but varying , the

component of the electric field is the component

ANSWER: =

ANSWER:

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=

Correct

Faraday's Law and Induced Emf Learning Goal: To understand the terms in Faraday's law and to be able to identify the magnitude and direction of induced emf. Faraday's law states that induced emf is directly proportional to the time rate of change of magnetic flux. Mathematically, it can be written as , where

is the emf induced in a closed loop, and

is the rate of change of the magnetic flux through a surface bounded by the loop. For uniform magnetic fields the magnetic flux given by .

, where

is the angle between the magnetic field

and the normal to the surface of area

To find the direction of the induced emf, one can use Lenz's law: The induced current's magnetic field opposes the change in the magnetic flux that induced the current. For example, if the magnetic flux through a loop increases, the induced magnetic field is directed opposite to the "parent" magnetic field, thus countering the increase in flux. If the flux decreases, the induced current's magnetic field has the same direction as the parent magnetic field, thus countering the decrease in flux. Recall that to relate the direction of the electric current and its magnetic field, you can use the right-hand rule: When the fingers on your right hand are curled in the direction of the current in a loop, your thumb gives the direction of the magnetic field generated by this current. In this problem, we will consider a rectangular loop of wire with sides exists as shown in . The resistance of the loop is

and

...