Magnetic inductance Solenoids - folk.uio.nofolk.uio.no/ravi/cutn/elec_mag/14_solenoid.pdf · 2015. 12. 21. · P.Ravindran, PHY041: Electricity & Magnetism 22 February 2013: Magnetic

Magnetic inductance & Solenoidsg

P.Ravindran, PHY041: Electricity & Magnetism 22 February 2013: Magnetic inductance, and Solenoid

Changing Magnetic Flux

• A changing magnetic flux in a wire loop induces an electric current.

• The induced current is always in a direction that opposes the change in flux.

These facts were discovered by Michael Faraday and represent a key connection between electricity and magnetism. One simple example of this is a magnet moving in and out of aelectricity and magnetism. One simple example of this is a magnet moving in and out of a wire loop. As a bar magnet approaches a wire loop along a line perpendicular to the loop, more and more field lines poke through the loop and the flux increases. To oppose this change in flux a current is induced in the direction shown.g

Note that the induced current produces its own magnetic field pointing to the right Also note that there is no battery

N S

right. Also note that there is no battery in the loop! This current will only exist when the flux inside the loop changes. When the magnet is withdrawn the flux

v decreases and current is induced in the other direction. There is no current when the magnet is still.


I

Induced emf’s and CurrentsThe current induced in a loop come not from a battery but from a changing magnetic flux We can think of the loop containing an imaginary batterymagnetic flux. We can think of the loop containing an imaginary battery that gets turned on whenever flux in the loop changes. The strength of this battery is called the emf (electromotive force); it’s symbol is a script , and it is measured in volts. The induced current is given by:

I = / R h R i th i t l i t i th lI = / R where R is the internal resistance in the loop.

itself depends on the rate at which the flux inside the loop is changing. itself depends on the rate at which the flux inside the loop is changing. If the flux is changing at a constant rate,

Ф /tThis is Faraday’s law of induction. The

The greater the change in flux the greater, the greater the induced emf,

= -ФB /t negative sign here indicates the emf opposes the change in flux.


The greater the change in flux the greater, the greater the induced emf, and greater the induced current.

Faraday’s Law (Maxwell’s III equation)Changing magnetic field gives rise to electric current.

Induced emf in the loop, due to changing magnetic flux. tB

E

i.e. rate of change of magnetic flux is the e.m.f. induced in the circuit.

If i h h k d h lΦB

If q0 is the charge taken around the loop. Then Force EqF 0Now work done in taking the charge around the loopNow work done in taking the charge around the loop will be lF ddW rEqW 20

rEqq 2 B (Magnetic field inward)rEqq 200

P

B dt

lE

td B

P

lE

S

B

P

dsBtt

d .lE tBEor

I t l f Diff ti l f


Integral form Differential form

Faraday's Law If ll l f i t th i ht th h ti fi ld (FiIf one pull a loop of wire to the right through a magnetic field (Fig.a). A current flow in the loop.

If one move the magnet to the left, holding the loop still (Fig. b).Again, a current flow in the loop.

With both the loop and the magnet at rest (Fig. c), if one changeth t th f th fi ld (b i th t i th il fthe strength of the field (by varying the current in the coil of anelectromagnet), current flow in the loop.


Changing B

The first experiment, is an example of motional emf, convenientlyexpressed by the flux rule.

But in the second experiment loop is stationary so force can not bemagnetic which is responsible for producing currentmagnetic which is responsible for producing current.Faraday thought:

A changing magnetic field induces an electric field.

the emf is again equal to the rate of change of the flux,

td B

P

lE

.B da


so E is related to the change in B by this equation

Last expression is the Faraday's law, in integral form.

By applying Stokes' theorem:

This is Faraday's law, in differential form

Note: in the static case (constant B) as Faraday's lawNote: in the static case (constant B) as Faraday s lawreduces to the old rule . 0E dl

or x E = 0

I E i 3 h i fi ld h f i l diffIn Experiment 3 the magnetic field changes for entirely differentreasons, but according to Faraday's law an electric field will beinduced giving rise to an emf -d/dt


induced, giving rise to an emf -d/dt.

For all three cases universal flux rule:

Whenever (due to any reason) the magnetic fluxthrough a loop changes, an emf will appear in theloop.

B

Summary:t

yIn Faraday's first experiment it's the Lorentz force lawat work; the emf is magnetic.; g

But in the other two it's an electric field (induced by theBut in the other two it s an electric field (induced by thechanging magnetic field) that does the job.


General form of Faraday’s Law

bU U

b aba b a

a

U UV V V E dsq

S th l t ti f d l d th iSo the electromotive force around a closed path is:

E ds E ds

And Faraday’s Law becomes:

BdE dsdt

dtA changing magnetic flux produces an electric field.


This electric field is necessarily non‐conservative.

Induced emf and Electric Fields

• An electric field is created in the conductor as a result of the changing magnetic flux

• Even in the absence of a conducting loop, a changing magnetic field will generate an electric field in empty spaceThi i d d l t i fi ld i ti• This induced electric field is nonconservative– Unlike the electric field produced by stationary chargescharges

• The emf for any closed path can be expressed as the line integral of E.ds over the pathintegral of E ds over the path


E produced by changing B

dBdE ddt

2d B

E 2 r r

E 2 r r

dt

d B

d BrE2 dt

2 dt


Faraday’s law: Changing magnetic field induces electrical current

• (a) When a magnet is moved• (a) When a magnet is movedtoward a loop of wire connectedto a galvanometer, thegalvanometer deflects as shown,indicating that a current isindicating that a current isinduced in the loop.

• (b) When the magnet is heldt ti th i i d dstationary, there is no inducedcurrent in the loop, even whenthe magnet is inside the loop.

• (c) When the magnet is movedaway from the loop, thegalvanometer deflects in theopposite direction, indicatingpp , gthat the induced current isopposite that shown in part (a).


Magnetic Force on a Current‐ Carrying Conductor

• For closed circuit of contour C carrying I , total magnetic force Fm is:

C

m dI N BlF

• In a uniform magnetic field, Fm is zero for a closed circuit

C

circuit.


Magnetic Force on a Current‐ Carrying Conductor

• On a line segment, Fm is proportional to the vector between the end points.

BF Im


Straight Wire PracticeDraw some magnetic field lines (loops in this case) along the wire.

I

Using x’s and dots to represent vectors into and out of the page show theUsing x s and dots to represent vectors into and out of the page, show the magnetic field for the same wire. Note B diminishes with distance from the wire.

. . . . . . . . . . . . B t f

I. . . . . . . . . . . . . . . . . . . . . . . .

B out of page

B into page


Current Loops and Magnetic FieldsThe magnetic field inside a current loop tends to be strong; outside it tends to be weakThe magnetic field inside a current loop tends to be strong; outside, it tends to be weak. Here’s why: Using the right hand rule we see that each length of wire contributes to a B field into the page (all lengths reinforcing one another). Outside the loop, say at P, the field is weak since the left side of the wire produces a field out of the page but the right sideweak since the left side of the wire produces a field out of the page, but the right side produces a field into the page.

Explain why the field is weak above the top wire.

The situation is the same with a circular loop. The effect is magnified with multiple turns of wire. Yet another right hand rule helps with current loops: Wrap your right hand in the direction of the loop and your thumb points in the direction of B inside. This is reminiscent

I

p y pof angular momentum for a spinning body.

I

IIPstrong field inside loop, strong field

into pagePdirected into

pageweak field outside

into page

weak field


Iweak field

Current Loops and Bar MagnetsNotice how similar the magnetic field of a current loop is to that of a simple bar magnet. Wrap your right hand along the loop in the direction of the g g pcurrent and your thumb points in the direction of the north pole of your electro‐magnet. Note also how the field lines are very close together inside the loop, just as they are when they thread through a bar magnet.

I


SolenoidSolenoidDistributed‐coiled conductorKey parameter: n loops/metre

BKey parameter: n loops/metre

If finite length, sum individual loops via B‐S LawI

If finite length, sum individual loops via S aw

If infinite length, apply Ampere’s Law B constant and axial inside, zero outsideRectangular path, axial length L

I

I II nBnLLB.d ovacovacenclovac BL

(use label Bvac to distinguish from core‐filled solenoids)Solenoid is to magnetostatics what capacitor is to electrostatics


g p

Solenoids• Solenoids are one of the most common• Solenoids are one of the most common electromagnets.• Solenoids consist of a tightly wrapped coil of wire, sometimes around an iron core. The multiple loops and the iron magnify the effect of the single loop g y g pelectromagnet. • A solenoid behaves as just like a simple bar magnet but only when current isbar magnet but only when current is flowing.• The greater the current and the more turns per unit length, the greater the field inside.• An ideal solenoid has a perfectly uniformAn ideal solenoid has a perfectly uniform magnetic field inside and zero field outside.


How Solenoids WorkThe cross section of a solenoid is shown. At point P inside the solenoid, the Bfield is a vector sum of the fields due to each section of wire. In the ideal case the magnetic field would be uniform inside and zero outside.

1 2 3 4 5 6 7 8

B = 0

I out of the page

1 2 3 4 5 6 7 8

PBx x x xx x x x

9 10 11 12 13 14 15 16I into the page

PB

9 10 11 12 13 14 15 16


Solenoids and Bar MagnetsA solenoid produces a magnetic field just like a simple bar magnet. Since it consists of many current loops, the resemblance to a bar y p ,magnet’s field is much better than that of a single current loop.


Magnetic Fields: Overview

Although the magnetic properties of electrons must ultimately be explained with h i hi k f i i i h h hquantum mechanics, we can think of magnetism arising whenever we have charge

in motion. This motion can be that of an electron (either spinning or orbiting) or it can be in the form of a current. Remember: moving charges produce magnetic fields, and external magnetic fields exert a magnetic force on moving charges (at least if the charge has a component of its velocity perpendicular to the field).



Applications of Ampere’s Law – A Solenoid

A solenoid is basically a bunch of loops of wire that are tightly wound. It is analogous to a capacitor which can produce a strong electric field. In this case it can produce a strong MAGNETIC FIELD.

Solenoids are important in engineering as they canSolenoids are important in engineering as they can convert electromagnetic energy into linear motion. All automobiles use what is called a “starter solenoid”. Inside this starter is a piston which is ppushed out after receiving a small amount of current from the car’s battery. This piston then completes a circuit between the car’s battery and starter motor yallowing the car to operate.


Applications of Ampere’s Law – A Solenoid

The first thing you must understand is what is the enclosed current. It is basically the current, I, times the # of turns you enclose, N.

When you integrate all of the small current elements they ADD up to the length of the solenoid, L

NILB

IdlB enc

)()( 0

0

iBLNnI

LNB

length per turns# ,

)()(

0

0It is important to understand that when you enclose a certain amount of turns that the magnetic field runs through the center of the solenoid. As a result the field lines and the niB osolenoid solenoid. As a result the field lines and the length of the solenoid are parallel. This is a requirement for Ampere’s Law.


ExampleExampleA solenoid has a length L =1.23 m and an inner diameter d =3.55 cm, and itA solenoid has a length L 1.23 m and an inner diameter d 3.55 cm, and it

carries a current of 5.57 A. It consists of 5 close‐packed layers, each with 850 turns along length L. What is the magnetic field at the center?

Nn lengthperturns#

niBL

n

osolenoid

length per turns#

B

xB )57.5(23.1

8505)1026.1( 6

0 02B 0.024 T


APPLICATIONS OF ELECTROMAGNETS: LIFTING MAGNET

http://www.youtube.com/watch?v=b6MVPYGveeQ&feature=related


Applications Of Electromagnets: Electric BellElectric Bell

http://www.youtube.com/watch?v=P96XG4Pa4OY


Applications Of Electromagnets: Relay•In this figure you can see that a relay •In this figure, you can see that a relay consists of two separate and completely independent circuits.

•The first is at the bottom and drives the electromagnet. In this circuit, a switch is controlling power to the electromagnet.

•When the switch is on, the electromagnet is on, and it attracts the armature (blue).

•The armature is acting as a switch in the second circuit.

•When the electromagnet is energized the •When the electromagnet is energized, the armature completes the second circuit and the light is on.

•When the electromagnet is not energized, the spring pulls the armature away and the circuit is not complete. In that case, the light is dark


is dark.

Documents

Magnetic inductance Solenoids - folk.uio.nofolk.uio.no/ravi/cutn/elec_mag/14_solenoid.pdf · 2015. 12. 21. · P.Ravindran, PHY041: Electricity & Magnetism 22 February 2013: Magnetic