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Chapter V Magnetostatics
Recommended problems: 5.1, 5.4, 5.8, 5.9, 5.10, 5.11, 5.13, 5.14, 5.15, 5.22,
5.23, 5.24, 5.27, 5.44, 5.45, 5.46.
The Lorentz Force Law
It is known that any stationary charge produces an e.f.
If a charge q is placed in an e.f. It will experiences an e. force according to
)1(EqF
Now a moving charge produces a m.f, B and any moving charge
in a magnetic field experiences a m. force according to.
LawLorentsBvqF )2(
From Eq.(2), it is clear that the m.force is perpendicular to both v & B.
Now since dt
sdv
0sdFm
The magnetic force doesn't do any work in a moving charge.
Motion of a Charged Particle in M. Field
It is shown that the magnetic force doesn’t do any work
If the magnetic force is the only force acting on a particle and knowing that
KWnet 0K
The speed is constant the acceleration is due to the change in direction
only the force is centripetal, i.e., the motion of the charge is circular.
Example 5.1 Describet the motion of a charged particle in a uniform m.field.
kBBLet ˆ
With B is constant.
2
2
dt
rdmBvqFNow m
jxBiyB
B
zyx
kji
BvBut ˆˆ
00
ˆˆˆ
kzjyixmjxBiyBq ˆˆˆˆˆ
Solution.
)3(yqBxm
)4(xqBym
)5(0zm
Integrating Eqs. 3-4 with respect to time, we get, respectively
)6(1CqByxm
)7(2CqBxym
)8(3Cz
Substituting for from Eq. (7) into Eq. (3) y
Cx
m
qBx
2
)9(22 axx m
qBwith
Solving Eq.(9) )10(cos otRax
Differentiating Eq. (10) w.r.t time and substituting into Eq. (6)
CytR o sin
)11(sin otRby
Squaring Eqs(10+11) and then adding we get
)12(222Rbyax
Then the path of the motion is a circle of radius R and centered at (a,b).
Now for the motion is helix with its axis in the direction of B. constantz
Now from Eqs.(10&11) we have
otRx sin otRy cos
Squaring the above two equations and then adding we get
22222 vwRyx qB
mvvR
Example 5.2 A particle of mass m and charge q starts from rest at the origin in
the presence of both E and B such that and . Describe the
motion of the particle.
kEE ˆ
iBB ˆ
Solution.
kyBjzB
B
zyx
kji
Bv ˆˆ
00
ˆˆˆ
BvqEqFFF me
)13(ˆˆˆ kyjzqBkqErm
)14(zqBym
)15(yqBqEzm
)16(0x
From Eq.(16), and since the particle stars from rest, then 0x
Integrating Eq.(14), and since the particle stars from rest, we get
)17(1 zCzy
Substituting Eq.(17) back into Eq.(15) we get
zm
qEz 2
m
qEzz 2
)18(cos otARz
Integrating Eq.(15), and since the particle stars from rest, we get
)19(2 ytm
qECyt
m
qEz
Differentiating Eq.(18) we get )20(sin otAz
Equating Eqs.(19 & 20) we obtain
ytm
qEtA o sin
)21(sin tm
qEtAy o
Squaring Eqs. (18 & 21) and then adding we get
)22(222RRztRy
Which is the equation of a circle of radius R and center positioned at (0,Rt, R)
travels along the y-axis at constant speed. The curve generated in this way is
called cycloid with.
)23(B
ER
)24(
B
ERv
Current
A moving charge in one direction produces current I with
)25(dvAnedt
dqI
With A is the cross sectional area of the wire, n is the No. of free charges per
unit volume, and vd is the drift velocity.
The magnetic force on current-carrying wire is
)26( BldIIdtBvdqBvFm
Example 5.3 A rectangular loop of wire, supporting
a mass m, hangs vertically with one end in a
uniform m.f. B. Find I such that m be in equilibrium.
Solution. For the mass to be in equilibrium we have
0gm FF
jmgFF gmˆ
jIaBkiIaBBlIFBut mˆ)ˆˆ(
IaBmgaB
mgI
The Current Density
The current density J is defined such that
)27( AdJI
Or from Eq.(25) we have
)28(dvJ
Now the magnetic force can be written as
)29( dBJdBvdqBvFm
Let us calculate the current enering a closed surface S. We have from Eq.(27)
S
dSnJI ˆ
The minus sign because n is outward.
Using the divergence theorem, we get
)30(ˆ dJdSnJIS
But the current is equal to the rate at which charge is transported into the
volume, i.e.,
)31(V
ddt
d
dt
dqI
Since the volume is constant and knowing that is a function of position as well
as of time, we get
)32(
Vd
tI
Equating Eqs. (30 & 32) and since the volume is arbitrary we get
)33(t
J
Continuity Equation
An outflow of current through a volume decreases the charge inside that
volume
The Biot-Savart Law
The m.f at a point due to a wire of steady current I is given by
)34(4 3
ldIB o
Is an element along the wire. ld
is a vector from the source to the point, and rr
27 /104 ANo is the magnetic permeability of free space.
dl
I
P
Example 5.5 Find the m.f a distance s
from a long straight wire carrying a
steady current I.
dz
z
I
P
s
Solution. We begin by identifying an
element of the wire, having length
dl=dz. Now
kdzld ˆ
ssr ˆ
kzr ˆ
kzssrr ˆˆ
33
ˆˆˆ
44
kzsskdzIldIB oo
2
12
3224
l
l
o
zs
dzIsB
22222 2
3Using
xaa
x
xa
dx
2
322
ˆ
4zs
dzIsB o
221
1
222
2
22 44
2
1
23
sl
l
sl
l
s
I
zs
z
s
IB o
l
l
o
If the wire in infinite we have 21 ll
s
I
sls
IlB oo
l
22lim
222
To find the force between two parallel wires we have for
the force on wire 1 due to wire 2
F1
a
I2 I1
B2
21112 BlIF
kd
IBBut o ˆ
2
22
is the m.f due to I2 at the position of I1
jld
IIF o ˆ
2
2112
The force is attractive if the currents are in the same direction
and repulsive if the currents are in opposite directions.
Example 5.6 Find the m.f due a distance z from the
center of a circular loop of radius R which carries a
steady current I.
Solution. We begin by identifying an element of the
wire, having length dl’ with
jiRdRdld ˆcosˆsinˆ
kzr ˆ jiRsRr ˆsinˆcosˆ
jRiRkzrr ˆsinˆcosˆ
zRR
dRdR
kji
ld
sincos
0cossin
ˆˆˆ
dkRjRziRz ˆˆsinˆcos 2
2
0 22
2
32
3
ˆˆsinˆcos
44zR
dkRjRziRzIldIB oo
dl’ I
P
z
R
0 yx BB 2
322
2
2 zR
IRB o
z
Note that at the origin of the loop (z=0 ) we get R
IB o
z2
Divergence and Curl of B
From Biot-Savart law we have
34
ldIB o
34
ldIB o
ABBABABut
ld
IldIB oo
33 44
0&1
3
But
ld
Ild
IB oo
1
4
1
4
)35(0 B
Again using Biot-Savart law we have
34
ldIB o
AdJI
&
)36(
4 3
d
rJB o
drJB o
34
BACCABCBAgU
sin
JJrJ
333
0&4Using3
J
drrrJB o
)37(rJB o
Ampere’s Law
Using Stoke's theorem we have
ldBSdB
rJB o
From Eq.(37) we have SdJldB o
)38(enco IldB
Where Ienc is the total current crossing the surface enclosed by the closed loop.
Like Gauss’s law, Ampere’e law is always true for steady current, but it is not
always useful. Only when the symmetry of the problem enables you to pull B
out of the integral , can you calculate B using Ampere’s law. ldB
The current configurations that can be handled by Ampere’s law are:
Infinite straight lines, infinite planes, infinite solenoids, toroids.
Example 5.9 Find the m.f a distance s from a long
straight wire carrying a steady current I.
Solution. For s>R, let us choose a circular loop
of radius s around the wire. Ampere’;s law is
R I
s
enco IldB
By symmetry we see that B must be constant in magnitude and
tangent to the loop at every point on the loop. Then
IsBdlB o )2(s
IB o
2
Which agree with that of example 5.5 in the limit of long wire.
For a point inside the wire the circular
loop in this case has a radius sR. R
I s
To find the current enclosed by the loop, we note that
22 s
I
R
IJ enc
2
2
R
sIIenc
Ampere’s law now gives
2
2
)2(R
sIIsBdlB enco 22 R
IsB o
At s=0, the magnetic field is zero, as expected
encoId lB
Example 5.8 Find the m.f of a very
long solenoid, consisting of n turns
closely wound turns per unit length on
a cylinder of radius R and carrying a
steady current I.
Solution. The m.field for a long
solenoid is zero outside and uniform
inside. To calculate the internal
magnetic field, we can apply Ampere’s
law to the rectangular loop abcd of
length l and width w.
enco IldB
enco
a
d
d
c
c
b
b
a
IldBldBldBldB
For the portions bc and da B is to dl and for the portion cd
B=0, then only the 1st integral survives
enco
b
a
IldB
For the portion ab B is to dl and its magnitude is constant
along it
enco IBl
But the current crossing the surface enclosed by the loop is I multiplied by the No. of turns bounded by the loop
nIIl
NBNIBl ooo
With n is the No. of turns per unit length.
Example 5.10 Find the m.f of
a toroid of steady current I.
Solution From the symmetry, the
magnetic field lines form
concentric circles inside the
toroid and there is no magnetic
filed outside. This means that B
is constant in magnitude along
the Amperian loop and directed
tangent to it. Therefore, at a point
inside the coil we have
enco IldB
r
NIBNIrB o
o
22
Magnetic Vector Potential
0B
From Eq.(35) we have
)39(AB
d
rJB o
34
13
But
Now from Eq.(36) we have
drJB o 1
4
AfAfAf
Using
rJrJrJ
1
Zero since J is
independent on r.
)40(
44
d
rJd
rJB oo
Comparing Eq.(39) with Eq.(40) we get
)41(
44
ldId
rJA oo
Example 5.11 A spherical shell of
radius R, carrying a uniform charge
density , is set spinning at angular
velocity . Find the vector potential it
produces at point r.
Solution: The current density is given
by Eq.(29)
vd
dSv
d
dqvJ
Sdvd
rJA oo
44
cossinsincossincos0sin
ˆˆˆ
RRR
kji
rvBut
k
jiR ˆsinsinsin
ˆsincoscoscossinˆcossinsin
krrthatKnowing ˆ
kRjRiRrRrand ˆcosˆsinsinˆcossinˆ
cos222 RrrRrr
0sincos2
0
2
0
ddasNow
The x-component and thee z-component of A cancel and we get
022
2
0
3
cos2
sincossin
4ˆ
RrrR
ddRjA o
rRRrrRrRRrrRr
RjA o 2222
2sin
6ˆ
RrRrRrrR
rRthatKnowing
RrR
Rrr
r
RjA o
3
3
2 2
2sin6
ˆ
jrr
kji
rBut ˆsin00
cos0sin
ˆˆˆ
Rrrr
R
RrrR
Ao
o
3
4
3
3
If is along the z-axis and r is arbitrary we have
The magnetic field can be now found from Eq.(39)
ˆsinˆcossinˆsinsin
cossinsincossin00
ˆˆˆ
rjrir
rrr
kji
r
Rrr
R
RrrR
Ao
o
2
4sin
3
ˆsin3
Example 5.12 Find the vector potential for an
infinite solenoid of n turns per unit length,
radius R and current I.
Solution: It is difficult to use Eq.(41) since the
wire is infinite. Defining the m.flux as
)42(SdB
SdA
)43( ldA
Using Stoke’s theorem we get
Taking a closed circular loop inside the solenoid of radius s<R and
noting that A is constant along that loop and tangent to it we get
sAsB 22
ˆ
2ˆ
2s
nIs
BA o
in
B=onI
For outside we take a circular loop of radius s>R so we have
sARB 22
ˆ2
ˆ2
22
s
RnIR
s
BA o
out
Use Eq.(39) to check if it gives the correct values of B.
znI
snI
zs
zss
sAB o
o
inin ˆ
02
0
ˆˆˆ
1
2
0
02
0
ˆˆˆ
1
2
nIR
zs
zss
sAB
o
outout
Multipole Expansion of the Vector Potential
Consider a circuit of current I. The
vector potential at a distant point r>>r` is
ldIA o
4
0
cos11
nn
n
Pr
r
r
But rom Eq.(29) of Chapter 3 we have
0
1cos
1
4 nn
n
no ldPr
r
IA
For monopole (n=0) we have
0
01
4 n
omonopole ld
r
IA
For dipole (n=1) we have
ldrrr
Ildr
r
IA oo
dipole
ˆ
1
4cos
1
4 22
SdrldrrBut
ˆˆ
2
ˆ
4 r
rmA o
dipole
AISdImWith
Is the magnetic dipole moment with its direction is determined by
the R.H.R.