Magnetostatics

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Chapter 5
Magnetostatics
Department of Physics , ROCMA
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• The Lorentz Force Law

Magnetic Fields
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• The Lorentz Force Law

Magnetic Fields
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• The Lorentz Force Law

Magnetic Forces
gt Lorentz Force
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• The Lorentz Force Law

Exp 1 cyclotron motion
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• The Lorentz Force Law

Exp 2 A more exotic trajectory occurs if we
include a uniform electric field, at
right angles to the magnetic one. Suppose, for
instance, that B points in the
x-direction, and E in the z-direction. A particle
at rest is released from the origin
what path will it follow?
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• The Lorentz Force Law

Exp 2
no force on x-direction
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• The Lorentz Force Law

Exp 2
Substitute to (1)
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• The Lorentz Force Law

Exp 2
The particle start from rest at origin
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• The Lorentz Force Law

Exp 2
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• The Lorentz Force Law

Current
the charge per unit time passing a given point
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• The Lorentz Force Law

Current
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• The Lorentz Force Law

Exp 3 A rectangular loop of wire, supporting a
mass m, hangs vertically with one end in a
uniform magnetic field B, which points into the
page in the shaded region of figure. For what
current I, in the loop, would the magnetic force
upward exactly balance the gravitational force
downward?
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• The Lorentz Force Law

Exp 3
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• The Lorentz Force Law

Surface current density
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• The Lorentz Force Law

Volume current density
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• The Lorentz Force Law

Summarize
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• The Lorentz Force Law

Exp 4(a) A current I is uniformly distributed
over a wire of circular cross section, with
radius a. Find the volume current density J .
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• The Lorentz Force Law

Exp 4(b) suppose the current density in the wire
is proportional to the distance from the axis.
Jks (kconstant) . Find the total current I in
the wire.
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• The Lorentz Force Law

Equation of continuity
total charge per unit time leaving a volume V
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• The Biot-Savart Law

The magnetic field of a steady current
Biot-Savart law
permeability of free space
units
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• The Biot-Savart Law

Exp 5 Find the magnetic field a distance s from
a long straight wire carrying a steady
current I .
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• The Biot-Savart Law

Exp 5
For an infinite wire
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• The Biot-Savart Law

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• The Biot-Savart Law

Exp 6 Find the magnetic field a distance z
above the center of a circular loop of radius
a, which carries a steady current I .
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• The Biot-Savart Law

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• The Divergence and Curl of B

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• The Divergence and Curl of B

Amperes law in integral form
from stokes theorem
Amperes law in differential form
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• The Divergence and Curl of B

from Biot-Savart law
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• The Divergence and Curl of B

since
The divergence of the magnetic field is zero
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• The Divergence and Curl of B

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• The Divergence and Curl of B

since
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• The Divergence and Curl of B

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• The Divergence and Curl of B

Amperes law
in differential form
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• Application of Amperes law

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• Comparison of Magnetostatics and Electrostatics

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• Application of Amperes law

Exp 7 Find the magnetic field a distance s from
a long straight wire, carrying a
steady current I.
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• Application of Amperes law

Exp 8 Find the magnetic field of an infinite
uniform surface current
, flowing over the xy plane.
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• Application of Amperes law

Exp 9 Find the magnetic field of a very long
solenoid, consisting of n closely
wound turns per unit length on a cylinder of
radius R and carrying a steady
current I .
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• Application of Amperes law

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• Application of Amperes law

Exp 9
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• Application of Amperes law

Exp 9
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• Application of Amperes law

Exp 10 A toroidal coil consists of a circular
ring, or donut, around which a
long wire is wrapped. The winding is uniform
and tight enough so that each
turn can be considered a closed loop. The
cross-sectional shape of the
coil is immaterial. I made it rectangular in
figure a for the sake of
simplicity, but it could just as well be circular
or even some weird
asymmetrical form, as in figure b, just as long
as the shape remains the same
all the way around the ring. In that case it
follows that the magnetic
field of the toroid is circumferential at all
points, both inside and
outside the coil.
(a)
(b)
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• Application of Amperes law

Exp 10
From Biot-Savart law
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• Application of Amperes law

Exp 10
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• Application of Amperes law

Exp 10
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• Application of Amperes law

Exp 10
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• Application of Amperes law

Exp 10
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• Magnetic Vector Potential

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• Magnetic Vector Potential

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• Magnetic dipole of the vector potential

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• Magnetic dipole of the vector potential

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• Magnetic dipole of the vector potential

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• Magnetic Vector Potential

Exp 11 A spherical shell, of radius D, carrying
a uniform surface charge s, is set spinning at
angular velocity ?. Find the vector potential it
produces at point r.
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• Magnetic Vector Potential

Exp 11
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• Magnetic Vector Potential

Exp 11
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• Magnetic Vector Potential

Exp 11
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• Magnetic Vector Potential

Exp 11
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• Magnetic Vector Potential

Exp 11
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• Magnetic Vector Potential

Exp 11
For D?r
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• Summary

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• Magnetostatic Boundary Conditions

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• Magnetostatic Boundary Conditions

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• Magnetostatic Boundary Conditions

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