Induction and Inductance

Induction Faraday’s Law of Induction Lenz’s Law Energy Transfer Induced Electric Fields Inductance and Self Inductance RL Circuits Energy Stored in a Magnetic Field Energy Density Mutual Induction pps by C Gliniewicz

When scientists first discovered that an electric current can create a magnetic field, they were surprised by the effect. Later when they discovered that a magnetic field can create a current , they were even more surprised. This second effect is called induction.

A current can be produced in a loop when a magnet is moved into or out of the loop. There has to be a relative motion of the magnet and loop. Either or both can be moving. The greater the relative motion, the greater the current that is produced. If the north pole of a magnet is moved toward a loop, the current may move clockwise and if the north pole is moved away the current moves opposite or counter-clockwise. Moving a south pole towards or away from a loop will produce a current moving opposite that of a north pole.

The current produced is an induced current and work done is called an induced emf. If two conducting loops are placed near one another and a current passes through one, a magnetic field is created. That magnetic field induces a current in the second loop.

Michael Faraday visualized this as magnetic field lines passing through the loop. The more lines, the greater the magnetic field, the greater the induced current.

pps by C Gliniewicz

When the number of lines intersecting the loop is changing, an induced emf is produced in the wire. If the magnetic field is unchanging, there is no emf induced.

Magnetic flux is defined is the same way as the electric flux.

If the magnetic field is perpendicular to the area, the flux is just the



A. The induced emf is just the time rate of change of the flux.

The magnetic flux through a coil can be changed by changing the magnetic field, or changing the area of the coil, or changing the angle between the coil and the magnetic field, or some combination of these.

Heinrich Fredrich Lenz found the rule for determining the direction of the induced current.

  

E 

d dt



dA

E  

N d

 

dt

Another way to visualize this is that the magnetic field of the induced current will point in the opposite direction of the magnetic field inducing the emf.

Electric guitar pickups operate using Lenz’s Law to change the motion of the steel strings into an electric signal.

If a conducting plate is moved through a magnetic field, the electrons move in a swirl inside the metal rather than in a set path. This swirling current is called a \n eddy current.

If one has a changing magnetic field, either increasing or decreasing, that changing field can create an electric field. A constant magnetic field causes no electric field. The equation for Faraday’s Law can be rewritten.

 pps by C Gliniewicz

ds



Lv

 

R P

 

d



dt F A Lx

   

Lv R

E

  

L

   

d





dt



R

 

2

R

    

Lv R

  

2

R

 

R



d dx



Lx

 

L dx dt

 

R v

 

Lv

 

R

Electric potential has meaning only for electric fields that are produced by static charges. There is no meaning for electric fields produced by induction.

Previously, one noted that that the closed integral of the electric field dotted with ds was zero. However this was with a static charge and no magnetic field. If one has a changing magnetic field the closed integral is the negative of the time rate of change of the magnetic flux.

An inductor is similar to a long solenoid which produces a magnetic field inside as the current changes. Inductance, L, is defined by the number of turns of the coil, the current and the magnetic flux.

Inductance is measured in Henries (H), tesla-metres squared per ampere. The value of the product of the number of turns multiplied by the magnetic flux, the numerator of the inductance is called the magnetic flux linkage.

The inductance per unit length is L L

pps by C Gliniewicz

l N

   

o



The permeability of free space can also be written in terms of the Henry.

An induced current appears in any coil where the current is changing.

Initially, an inductor acts to oppose changes in the current through it. After a long time, an inductor acts like an ordinary connecting wire.

We can have a circuit with a resistance and an inductor, an RL circuit. The potential around an RL circuit must add to zero just like any circuit. One can write equation in the same manner as in an RC circuit.

This describes the current as it rises. When the current falls after being turned off, it is described by



L o

 



pps by C Gliniewicz



L R d e







t

 

L N R



7 o e



0 A



L L



d dt

 E

  

10 L d 7 dt



H m R

 

E

R

  

1



e



Rt L

  



L



L R

Energy can be stored in an inductor. Energy density is the amount of energy in the inductor per unit volume of the inductor.

One coil of wire can cause a self inductance because of the changing magnetic field. When there are two coils, the changing field in one coil induces an emf in the second when creates a magnetic field which is changing inducing a magnetic field in the first coil. This is called mutual induction.

E E

2 2 U



2

pps by C Gliniewicz  

on



1 1 2



1 L L 2 2

 

1 2 on u

  

M

 

M 2 on 1 d dt



1 d dt



1



and and L



2 2 M 2



L on

 E

1

E

1



2 A

   

M d 2 2



2

 

2 o on 1 M dt



2 1 on 2 d dt



2 M 2 on 1 d dt



1



N 2 d



2 on dt 1 M 2 on 1



M 1 on 2



M