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.
P.Ravindran, PHY041: Electricity & Magnetism 5 February 2013: Magnetic dipoles, inductance, and torque
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
P.Ravindran, PHY041: Electricity & Magnetism 5 February 2013: Magnetic dipoles, inductance, and torque
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
P.Ravindran, PHY041: Electricity & Magnetism 5 February 2013: Magnetic dipoles, inductance, and torque
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.
P.Ravindran, PHY041: Electricity & Magnetism 5 February 2013: Magnetic dipoles, inductance, and torque
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
P.Ravindran, PHY041: Electricity & Magnetism 5 February 2013: Magnetic dipoles, inductance, and torque
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
P.Ravindran, PHY041: Electricity & Magnetism 5 February 2013: Magnetic dipoles, inductance, and torque
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
P.Ravindran, PHY041: Electricity & Magnetism 5 February 2013: Magnetic dipoles, inductance, and torque
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:
P.Ravindran, PHY041: Electricity & Magnetism 5 February 2013: Magnetic dipoles, inductance, and torque
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
P.Ravindran, PHY041: Electricity & Magnetism 5 February 2013: Magnetic dipoles, inductance, and torque
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.
P.Ravindran, PHY041: Electricity & Magnetism 5 February 2013: Magnetic dipoles, inductance, and torque
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'),
P.Ravindran, PHY041: Electricity & Magnetism 5 February 2013: Magnetic dipoles, inductance, and torque
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
P.Ravindran, PHY041: Electricity & Magnetism 5 February 2013: Magnetic dipoles, inductance, and torque
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
P.Ravindran, PHY041: Electricity & Magnetism 5 February 2013: Magnetic dipoles, inductance, and torque
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
P.Ravindran, PHY041: Electricity & Magnetism 5 February 2013: Magnetic dipoles, inductance, and torque
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
P.Ravindran, PHY041: Electricity & Magnetism 5 February 2013: Magnetic dipoles, inductance, and torque
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.
P.Ravindran, PHY041: Electricity & Magnetism 5 February 2013: Magnetic dipoles, inductance, and torque
• 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)
P.Ravindran, PHY041: Electricity & Magnetism 5 February 2013: Magnetic dipoles, inductance, and torque
Circular current loop
B
dBp2
0 RIB 2/322
0
2 zRB
rz
R
P.Ravindran, PHY041: Electricity & Magnetism 5 February 2013: Magnetic dipoles, inductance, and torque
Circular current loop
20 RIB
2/3220
2 zRB
If z >>R
20 RIB
32 zB
2
3
20
2 zRIB
P.Ravindran, PHY041: Electricity & Magnetism 5 February 2013: Magnetic dipoles, inductance, and torque
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.
P.Ravindran, PHY041: Electricity & Magnetism 5 February 2013: Magnetic dipoles, inductance, and torque
Magnetic dipole in an external Magnetic field
Magnetic dipole experiences a torque
P.Ravindran, PHY041: Electricity & Magnetism 5 February 2013: Magnetic dipoles, inductance, and 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.
P.Ravindran, PHY041: Electricity & Magnetism 5 February 2013: Magnetic dipoles, inductance, and torque
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).
P.Ravindran, PHY041: Electricity & Magnetism 5 February 2013: Magnetic dipoles, inductance, and torque
n̂nB
zB
e
y yb
f)ˆ()90sin( xIbBF
P.Ravindran, PHY041: Electricity & Magnetism 5 February 2013: Magnetic dipoles, inductance, and torque
)()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
P.Ravindran, PHY041: Electricity & Magnetism 5 February 2013: Magnetic dipoles, inductance, and torque
)( 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
P.Ravindran, PHY041: Electricity & Magnetism 5 February 2013: Magnetic dipoles, inductance, and torque
q
Fn̂Fr
z xbIaBR ˆsin2
e F
2
xbIaB ˆsin Fhe
y
xIaBL sin2
F
y
fFfg
P.Ravindran, PHY041: Electricity & Magnetism 5 February 2013: Magnetic dipoles, inductance, and torque
Torque on a current loop
xIabB ˆsin xIabBsin
xIAB ˆsin
)ˆ( BNIA)( BnNIA
P.Ravindran, PHY041: Electricity & Magnetism 5 February 2013: Magnetic dipoles, inductance, and torque
Torque of a magnetic dipoleq g p
BnIA
ˆ BnIA
B
Torque tends to rotate so that it lines up with B.
P.Ravindran, PHY041: Electricity & Magnetism 5 February 2013: Magnetic dipoles, inductance, and torque
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
P.Ravindran, PHY041: Electricity & Magnetism 5 February 2013: Magnetic dipoles, inductance, and torque
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.
P.Ravindran, PHY041: Electricity & Magnetism 5 February 2013: Magnetic dipoles, inductance, and torque
Find the magnetic dipole moment of the circuit.
AiA
22 ba 2
bai
2
P.Ravindran, PHY041: Electricity & Magnetism 5 February 2013: Magnetic dipoles, inductance, and torque
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.
P.Ravindran, PHY041: Electricity & Magnetism 5 February 2013: Magnetic dipoles, inductance, and torque
xzIayIa
The field of aThe field of a i di lmagnetic dipole
P.Ravindran, PHY041: Electricity & Magnetism 5 February 2013: Magnetic dipoles, inductance, and torque
Electric field of an electric dipolep
P.Ravindran, PHY041: Electricity & Magnetism 5 February 2013: Magnetic dipoles, inductance, and torque
Magnetic field lines of a magnetic g gdipole
P.Ravindran, PHY041: Electricity & Magnetism 5 February 2013: Magnetic dipoles, inductance, and torque
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.
P.Ravindran, PHY041: Electricity & Magnetism 5 February 2013: Magnetic dipoles, inductance, and torque
Similarities
Electric and magnetic dipole fields vary as r‐3 when we are far from the dipoles.
P.Ravindran, PHY041: Electricity & Magnetism 5 February 2013: Magnetic dipoles, inductance, and torque
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.
P.Ravindran, PHY041: Electricity & Magnetism 5 February 2013: Magnetic dipoles, inductance, and torque