magnetic dipp,oles, inductance -...

Magnetic dipoles, inductance & Torqueg p , q

P.Ravindran, PHY041: Electricity & Magnetism 5 February 2013: Magnetic dipoles, inductance, and torque

Stationary Charges ➾ Constant electric fields: electrostatics.Steady Currents ➾ Constant magnetic fields: magnetostaticsSteady Currents ➾ Constant magnetic fields: magnetostatics.

Static electric fields characterized by E or D and related according to D = εE.

Static magnetic fields characterized by H or B and relatedStatic magnetic fields characterized by H or B and related according to B = μH.

Th t j l i t t tiThere are two major laws governing magnetostatics:1. Bio-Savart law2. Ampere’s circuital law2. Ampere s circuital law

Just as Gauss’s law is special case of Coulombs law, Ampere’s law is a special case of Biot-Savart’s law and is easily applied in problems involving symmetrical current distribution.


PROBLEM

If a magnetic field, H = 3ax + 2ay A/m exists at a point in free space, what isthe magnetic flux density at the point ?

Solution:H = 3 ax + 2 ay

0 = permeability of free space

= 4 x 10-7 H/m4 x 10 H/m

B = H (for free space =0 )

= 4 x 10-7(3ax + 2ay)

= (3 767 a + 2 513 a ) x 10-6(3.767 ax + 2.513 ay) x 10

B = (3.767 ax + 2.513 ay) wb/m2


The Curl of B (Ampere’s law) Let’s assume an infinite straight wire through whichcurrent is coming out of the page.

The integral of B around a circular path of radius s,d h i icentered at the wire, is


Now suppose we have a bundle ofstraight wires.

Each wire that passes through ourl t ib t I d thloop contributes 0I, and thoseoutside contribute nothing.

The line integral will then be

where Ienc stands for the total current enclosed by the integrationenc y gpath. If the flow of charge is represented by a volume currentdensity J, the enclosed current is

the integral taken over the surfacebounded by the loop.


So, we get:

Applying Stokes' theorem (line integral to surface integral):

The equation for the curl of B is called Ampere's law(i diff ti l f )(in differential form).

This is the integral version of Ampere's lawThis is the integral version of Ampere s law

For currents with appropriate symmetry, Ampere's law in integral form can beused easily for calculating the magnetic field


used easily for calculating the magnetic field.

Ampere’s Circuital Law in Integral Form

• Ampere’s Circuital Law in integral form states that “the circulation of the magnetic flux density in free space is proportional to the total current through the surfaceproportional to the total current through the surface bounding the path over which the circulation is computed.”

enclC

Id 0 lB

The line integral is around any closed contour bounding an open surface S.I is current through S:Iencl is current through S:

encl dI SJS

Where J is defined as current density


Ampere’s Circuital Law in Differential form

enclosedId 0lB

encl ddI SBlB0Applying Stroke's theorem to left‐hand side of above equation

L S

encl dI SJBut S

JB

Comparing the surface integrals in above expressions

JB 0

Since

This shows that magnetostatic field is non conservative in nature

0 JH


This shows that magnetostatic field is non conservative in nature.

Note: Conservative or non conservative field?

A force field F is called conservative (i.e. it is a conservative vector field – path i d d t) if it t f th i l t diti

:1 The curl of F is zero:

independent) if it meets any of these equivalent conditions:

1. The curl of F is zero:

2 The work W is zero for any simple closed path:2. The work, W, is zero for any simple closed path:

Th f b i h di f i l ΦThe force can be written as the gradient of a potential, Φ:


PROBLEM

If H is given by H = y cos 2x ax + (y + ex) az, determine J atthe origin.g

Differential form of Ampere’s circuit law isSolution:

x H = J

zyx aaa

xeyxyzyx

02cos

xyy

axyz

eyx

aeyy

a zx

yx

x 2cos2cos yy

J = ax – ex ay – cos 2x az

At (0 0 0) J = (a – a – a ) A/m2


At (0, 0, 0) J = (ax ay az) A/m

Applying Ampere’s Law

enclosedQdSE lId 0 lB

• Select a surface • Select a path

0E dSE encl

CId 0 lB

Select a surface– Try to imagine a surface where

the electric field is constant everywhere. This is accomplished if the surface is equidistant from

Select a path– Try to imagine a path where

the magnetic field is constant everywhere. This is accomplished if the surface isif the surface is equidistant from

the charge.– Try to find a surface such that

the electric field and the normal

accomplished if the surface is equidistant from the charge.

– Try to find a path such that the magnetic field and the path

h d lto the surface are either perpendicular or parallel.

• Determine the charge inside the surface

are either perpendicular or parallel.

• Determine the current inside the surfacesurface

• If necessary, break the integral up into pieces and sum the results.

surface• If necessary, break the integral up

into pieces and sum the results.


The Divergence of B

The Biot-Savart law for a volume current

This formula gives the magnetic field at a point r = (x y z) inThis formula gives the magnetic field at a point r = (x, y, z) in terms of an integral over the current distribution J(x', y', z').

B is a function of (x,y,z),B is a function of (x,y,z),

J is a function of (x', y', z'),


Here the integration is over the primed coordinates.The curl are to be taken with respect to the unprimed coordinates because B isp pa function of (x, y, z).

Applying the divergence to above equation,

, because J doesn't depend on the unprimed variables (x, y, z)and

S M l t i t


So Monopole can not exist.

The divergence and curl of the electrostatic field are:

These are Maxwell's equations for electrostatics.

The divergence and curl of the magnetostatic field are:

These are Maxwell's equations for magnetostatics


These are Maxwell s equations for magnetostatics.

M i iMagnetic properties

Magnetic dipoles

Field of a magnetic dipole

Force on a dipole in a non uniform field

Induced magnetic dipole momentInduced magnetic dipole moment


Problem: Magnetic field around an infinitely long wirelong wire

But B has the same value a distance aaway from the rod and hence y

enclId 0 lB


C

Biot‐Savart Law• Currents, i.e. moving electric charges, produce magnetic

fields. There are no magnetic charges• The Biot‐Savart Law relates magnetic fields to the currents

which are their sources. In a similar manner, Coulomb's law relates electric fields to the point charges which are their sources.


• where µ0 is the permeability constant

The Biot-Savart Law is used to calculate the magnetic field at a given position. It is usually only practical to do the calculation in special cases where some symmetry makes the problem simpler.

• What is the magnitude of the magnetic fieldat the center of a loop of radius R, carrying

I?

I

current I?

Idxr

R(a) B = 0 (b) B = (µ0I)/(2R) (c) B = (µ0I)/(2pR)

To calculate the magnetic field at the center we must use theTo calculate the magnetic field at the center, we must use theBiot-Savart Law:

Two nice things makes our calculation very easy:

Idx is always perpendicular to rr is a constant (=R)r is a constant ( R)


Circular current loop

B

dBp2

0 RIB 2/322

0

2 zRB

rz

R


Circular current loop

20 RIB

2/3220

2 zRB

If z >>R

20 RIB

32 zB

2

3

20

2 zRIB


2 z

0B

30

2 zB

2 zWh i R2 IWhere is = R2 I

d B i h di i

Known as Magnetic dipole moment• and B are in the same direction.

• B is produced by P.Ravindran, PHY041: Electricity & Magnetism 5 February 2013: Magnetic dipoles, inductance, and torque

• B is produced by .

• In general = I A• In general, = I A, true for any shape of the loop.

• A is a vector area,,with direction assignb right hand r leby right hand rule.


Magnetic dipole in an external Magnetic field

Magnetic dipole experiences a torque


Torques and Forces on Magnetic Dipoles in a uniform magnetic fieldDipoles in a uniform magnetic field

A magnetic dipole experiences a torque in a magnetic fieldA magnetic dipole experiences a torque in a magnetic field.

Let's calculate the torque ona rectangular current loopg pin a uniform field B.

Any current loop could bebuilt up from infinitesimalrectanglesrectangles.


Center the loop at the origin, and tilt it an angle from the z axis towards the y axis. y

Let B point in the z direction. The forces on the two vertical sides cancel (they tend tostretch the loop, but they don't rotate it).


n̂nB

zB

e

y yb

f)ˆ()90sin( xIbBF


)()90sin( xIbBFef

n̂B

zh

y

g y

bg

)ˆ()90sin( xIbBF P.Ravindran, PHY041: Electricity & Magnetism 5 February 2013: Magnetic dipoles, inductance, and torque

)()90sin( xIbBFgh

Forces on a current looppz n

h

ye

g

b a

x f

IaBF IaBF y

)( directiony

IaBF fg

)( directionyIaBFhe


)( directiony )( directiony

Forces on a current loop

)90i (I BF )90i (I BF

)(

)90sin(

di ti

IaBFef )(

)90sin(

directionx

IaBFgh

)( directionx )( directionx

IaBFfg IaBFhe

)( directiony )( directiony

Contribute to torque


q

Fn̂Fr

z xbIaBR ˆsin2

e F

2

xbIaB ˆsin Fhe

y

xIaBL sin2

F

y

fFfg


Torque on a current loop

xIabB ˆsin xIabBsin

xIAB ˆsin

)ˆ( BNIA)( BnNIA


Torque of a magnetic dipoleq g p

BnIA

ˆ BnIA

B

Torque tends to rotate so that it lines up with B.


Potential energy for the dipole

• B makes an angle ith the dipole• B makes an angle with the dipole

dU

BU

U has smallest value when andand BB areare parallelparallel ‐‐ BB

Largest when antiLargest when anti‐‐parallel parallel BB


Important to notep

• In general, potential energy (PE) can not be defined for a f ld lMagnetic field alone.

Si t th di l d d th it iti• Since torque on the dipole depends upon the its positionwith respect to the field, PE can be defined for magneticdipole in the field.p

• This PE corresponds to any change in the rotational configuration.


Find the magnetic dipole moment of the circuit.

AiA

22 ba 2

bai

2


z Find the magnetic dipole moment of the loop Allmoment of the loop. All sides have equal length aand it carries a current Iand it carries a current I.

a

ya ya

aDi ti i la

zIayIa ˆˆ 22 Direction is along the line y = z.


xzIayIa

The field of aThe field of a i di lmagnetic dipole


Electric field of an electric dipolep


Magnetic field lines of a magnetic g gdipole


Bar magnet can also be considered to be a magnetic dipolemagnetic dipole.

Field lines do not startor end but continueth h th i t i fthrough the interior ofthe magnets, formingclose loops.close loops.


Similarities

Electric and magnetic dipole fields vary as r‐3 when we are far from the dipoles.


p

Force on a dipole in a nonuniform fieldForce on a dipole in a nonuniform field

In a uniform field total force on the dipole (electricIn a uniform field total force on the dipole (electric as well as magnetic) is zero. There is only torque but no net motionno net motion

In a non uniform field net force is not zero. Dipole may move.may move.


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