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Electromagnetic Induction

©2011 by Bryan Pflueger

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Induced CurrentsIf we have a galvanometer attached to a coil of wire we can induce a current simply by changing the magnetic field passing through the coil.

To generate the magnetic field we can use either a bar magnet or an electromagnet, and we can vary its strength by moving it closer or further away from the coil. When the magnetic field increases within the coil, the induced current flows in some direction, and when the magnetic field decreases the current flows in the opposite direction.

When the bar magnet or electromagnet is held stationary above the coil there is no induced current because it is the result of a fluctuating magnetic field.

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1 In each of the following situations, a bar magnet is aligned along the axis of a conducting loop. The magnet and the loop move with the indicated velocities. In which situation will the bar magnet NOT induce a current in the conducting loop?

A B

C D E

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Magnetic FluxBefore we begin to explain how a current is induced and in which direction it flows, we will first discuss magnetic flux.

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2 A wire loop of area A is placed in a time-varying but spatially uniform magnetic field that is perpendicular to the plane of the loop, as shown above. The induced emf in the loop is given by ε = bAt1/2, where b is a constant. The time -varying magnetic field could be given by

A

B

C

D

E

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#

t

3 A loop of wire is placed in a constant magnetic field B and begins to rotate about its diameter. Which of the following represents the magnetic flux through the loop as a function of time?

#

t

A #

t

B

C D #

t

E#

t

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Faraday's LawFaraday's Law of Induction:

To determine the direction of the induced emf and current:· Define a direction for vector area A.· Determine in which direction the magnetic field is increasing or decreasing· Determine the sign of the magnetic flux, when the magnetic flux is positive the current is negative, but when the magnetic flux is negative the current is positive.· Use the first right hand rule, extend your thumb in the direction of vector area A, if the current is positive when you wrap your fingers around they are in the direction of the induced current, but if it is negative then it is in the opposite direction.

(For more then one loop)

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Lenz's LawLenz's Law is derived from Faradays Law of Induction and simply states that the induced current opposes any change in the magnetic flux through the system.

Lenz's law gives us only the direction that the induced current flows, it does not give us a means of calculating the magnitude of the current that will oppose the change. The current however can be determined by knowing the resistance of the material through which the magnetic field is passing through.

If the resistance is greater, there would be less current to oppose the change in the magnetic field and if the resistance were less there would be a greater current to oppose the change in the magnetic field.

In some cases with superconductors the resistance is zero, so the induced emf will result in a current, but the current will still remain even after the induced emf has dissipated.

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Motional Electromotive Force

v

+

-

A wire of length L is moving through a magnetic field with a velocity of v. As it moves through the magnetic field the free charges move to opposite sides, forming one positive and one negative end, creating a potential difference in the rod.

The potential difference can be written as:

Potential difference between plates generating a uniform electric field

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4 A square wire loop with side length l travels with speed v through a magnetic field of strength B. Which of the following will not induce a current in the loop?

A increasing the strength of the magnetic field

B stretching the loop making a larger circle

C moving the loop parallel to the field

D removing the loop from field

E rotating the loop about a diameter

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5 A square wire loop with side length l travels with speed v through a magnetic field of strength B. Which of the following represents the magnetic flux through the loop?

A BlvB Bl2

C zeroD Bv/l2

E B2v

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Induced Electric FieldA loop of wire with a galvanometer attached to it has a solenoid with a time varying magnetic field placed in it. During this time the magnetic field induces an emf in the wire, as well as a current. But think about this for a minute. How is it possible that charges are flowing when the magnetic force is dependent on the velocity of the loop and the loop is stationary?

The answer is an electric field is induced from a changing magnetic flux, and this electric field is different then what we are used to dealing with. It is not conservative, meaning after the particle has made one trip around the loop the net work done on it must equal q#.

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Induced Electric Field

The magnetic flux through the loop of wire is given by:

The line integral of the Electric Field is equal to the emf:

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Eddy CurrentsA metal disk is rotated about the axis perpendicular to its flat surface and a magnetic field is confined to a small circle which falls on the disk. The changing magnetic field creates an electric field which flows through the crystalline structure of the conductor and results in a current looping back in on itself. The charges flowing through the magnetic field create a force opposing the rotation of the disk.

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Displacement Current

When charging a capacitor we run into a problem with ampere's law. The current, IC, flowing through the surface of the conductor is zero, while the surface integral of the magnetic field with respect to the length is also zero.

While both of these values are zero, the electric field and the electric flux are increasing through the surface and we can derive equations for their values.

(Displacement Current)

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Displacement Current Density

The current Density equation is the same as that of a normal electric current. It can also be represented by the changing electric field between the capacitors.

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Displacement Current DensityTo test if the displacement current actually has any meaning, we will look at the simple case of a parallel plate capacitor, whose plates are circular and have a radius of R.

If what we found out to be true before then, at a radius r, r<R, there will be a magnetic field because there is an electric field.

The total current enclosed within this area can be represented by the displacement current density times the area for the smaller radius. But since the displacement current is the same as the conduction current for the charging capacitor we can write Ampere's Law as:

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Displacement Current Density

The equation we found holds true when we experimentally determine the value of the magnetic field, and for r>R the magnetic field is like that of a current carrying wire. This shows that the displacement current actually is a source of the magnetic field.

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Maxwell's EquationsMaxwell's equations are a collection of 4 different equations which show the relationship between electric fields and magnetic fields. This is not all Maxwell's work, but like Newton he is credited for discovering the relationship and making it clear to understand.

Gauss's Law: Shows how the enclosed charge produces an electric field

Gauss's Law for Magnetic Field: States that monopoles do not exist

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Ampere's Law with the displacement current: Shows that magnetic fields can be generated by a changing electric field

Maxwell's Equations

Faraday's Law: Shows that an electric field can be generated from a changing magnetic flux

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Maxwell's EquationsIf we tried to apply these laws to a case where we were in empty space, meaning there is no enclosed charge or the conduction current is zero, we could find other forms of the equations which apply for this case.

To do this we have to represent the magnetic and electric flux as:

If we were to add up all the forces acting on a particle, the electric and the magnetic we would see:

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SuperconductivityA superconductor is a material which if cooled to low enough temperatures loses all of its electrical resistance. But we will discuss two unique cases involving a magnetic field.

In order to achieve superconductivity the temperature has to drop below the critical temperature, TC. After this happens if a magnetic field is passed through the material and slowly increases it begins neutralize its superconductivity. The magnetic field required to eliminate superconductivity for a temperature less then TC is referred to as the critical field, BC.

Instead we will now pass a magnetic field through the material and then cool it down to where the material reaches superconductivity. When this happens the magnetic field lines that were originally passing through it are distorted and the field inside is now zero.

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