Magnetostatics

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Straight line currents
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Straight line currents
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we have a bundle of straight wires and some of
the wires passes through the loop
I5
I2
I3
I1
Loop
I4
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For surface current and volume currents
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for any arbitrary current distribution
The Divergence and Curl of B
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Applications of Amperes Law

• in
• Straight wire
• Sheet of Current
• Long Solenoid
• Toriodal Coil

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Straight Wire
Amperian Loop
s
I
B
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Infinite Sheet of Current
z
y
x
Amperian Loop
x
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Long Solenoid
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Problem 5.13 A steady current I flows down a long
cylindrical wire of radius a. Find the magnetic
field, both inside and outside the wire, if(a)
The current is uniformly distributed over the
outside surface of the wire.(b) The current is
distributed in such a way that J is proportional
to s, the distance from the axis.
I
a
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Problem 5.15 Two long coaxial solenoids each
carry current I, but in opposite directions as
shown. The inner solenoid (radius a ) has n1
turns per unit length, and the outer one(radius
b) has n2 turns per unit length.
b
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Problem A very long straight conductor has a
circular cross-section of radius R and carries a
current Iin. Inside the conductor there is a
cylindrical hole of radius a whose axis is
parallel to the axis of the conductor and a
distance b from it.
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find the magnetic field B at a point (a) on the
x axis at x2R and (b) on the y axis at y2R.
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(c) at a point inside the hole.
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Problem 5.14 A thick slab extending from z-a to
za carries a uniform volume current ,
Find the magnetic field, as a function of z,
both inside and outside the slab.
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Problem5.16 A large parallel plate capacitor
with uniform surface charge density s on the
upper plate and s on the lower is moving with a
constant speed v, as shown below.
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Find, (a) The Magnetic field between the plates,
(b) The Magnetic force per unit area on the
upper plate its direction,(c) the speed v at
which the magnetic force balances the electrical
force.
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Toriodal Coil
a circular ring, or donut around which
along wire is wrapped. ..the winding is uniform
and tight enough so that each turn can be
considered a closed loop.
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Problem the magnetic flux through the end face
of a solenoid
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Comparison of Magnetostatics and Electrostatics
The divergence and curl of the electrostatic
field are
together with the Boundary conditions
determine the field uniquely.
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Comparison of Magnetostatics and Electrostatics
The divergence and curl of the Magnetostatics
field are
together with the Boundary conditions determine
the field uniquely.
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Magnetic Vector Potential
permits us to introduce a Vector Potential A
in Magnetostatics
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Problem A spherical shell, of radius R, carrying
a uniform surface charge s, is set spinning at
angular velocity ?. Find the vector potential A
it produces at point P.
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Expressions for the Magnetic Field Inside
Outside the Spherical Shell are
(Inside the Spherical Shell)
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Expressions for the Magnetic Field Inside
Outside the Spherical Shell are
(Outside the Spherical Shell)
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Problem 5.42 Calculate the Magnetic Force of
Attraction between the Northern and Southern
Hemispheres of a Spinning Charged Spherical
Shell(of Radius R carrying a uniform charge
density s and spinning at an angular velocity ?).
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A Spinning Shell
(rR at the surface)
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Problem Find the vector potential of an infinite
solenoid with n turns per unit length, radius R,
and current I.
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Problem5.22 Find the magnetic vector potential
of a finite segment of straight wire, carrying a
current I.
z2
z1
o
I
z
s
rs
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Problem 5.24 If B is uniform, show that,

works.
Is this result unique, or there are other
functions with same divergence and curl.
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Here, r is the projection of vector-r on
x-y plane.
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Problem 5.23 What current density would produce
the vector potential,
(where k is a constant), in cylindrical
coordinates ?
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Multipole Expansion of the Vector Potential
P
r
rs
O
r/
dr/dl/
I
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Problem 5.60 (a) Work out the Multipole
expansion for the vector potential for a volume
current J.(b) Write down the Monopole potential
and prove that it vanishes.(c) Write the
corresponding dipole moment m.
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Problem Find the magnetic dipole moment of the
bookend-shaped loop as shown below. All sides
have length w, and it carries a current I.
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The Magnetic Field of a Pure Dipole
x
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Problem5.34 Show that the magnetic field of a
dipole can be written in coordinate free form as
x
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Problem 5.34 A circular loop of wire, with
radius R, lies in the xy-plane, centered at the
origin, and carries current I running
counterclockwise as viewed from the positive
z-axis.
z
R
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(a) What is its magnetic dipole
moment?(b) What is the (approximate) magnetic
field at points far from the origin? (c) Show
that, for points on the z-axis, the answer is
consistent with the exact field when zgtgtR.
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Problem5.35 A phonograph record of radius R,
carrying a uniform surface charge s, is rotating
at constant angular velocity ?. Find its dipole
moment.
z
?
0
R
y
x
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Fields due to Pure dipole Physical dipole.
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Problem 5.55 A Magnetic dipole
is situated at the origin, in an
otherwise uniform magnetic field
Show that there exists a spherical surface,
centered at the origin, through which no
Magnetic field line passes.
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Magnetostatic Boundary Conditions
Babove-
K
Bbelow-
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Magnetostatic Boundary Conditions
BIIabove
K
BIIbelow
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Thus The Perpendicular Component of the
Magnetic Field is Continuous across a surface
current.Whereas The Component of B that is
parallel to the surface but perpendicular to
the current is discontinuous by an amount µ0K.
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These Results can be Summarized as
where, n cap is the unit Vector perpendicular
to the Surface pointing upwards.
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Problem Show that the Vector Potential A is
continuous, but its derivative inherits a
discontinuity across any boundary.
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Problem 5.24 (a)find the vector potential a
distance s from an infinite straight wire
carrying a current I. (b) find the vector
potential inside the wire if it has a radius R
and the current is uniformly distributed.
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J
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END OF THE MAGNETOSTATICS