Electrostatics in Vacuum, in Conductors and in the
Presence of Linear Dielectrics
Principle of Superposition
Charges were at rest
0
E
Magnetostatics
• Look into the Forces between
the charges which are in motion
• ….the Types of Current distributions
• Continuity Equation
• Magnetic Field of a Steady Current
• The Divergence and Curl of B
• Magnetic Vector Potential A
The Forces between the charges which are in motion
(a) Current in opposite direction
(b) Current in same direction
What we are encountering is:The Magnetic Force
• Other Example is :
Magnetic Compass
….. the Needle will point towards the direction of the local magnetic field
…..for instance towards the Geographic North.
What if it is in the vicinity of a current carrying wire??
…trajectory of a Charged particle in the presence of an Uniform Electric field which is at Right
angles to a Magnetic Field.
x
y
z
E
B
Problem 5.4: Suppose that the magnetic field in some region has the
form
Find the force on a square loop, lying in the y-z plane, if it carries a
current I, flowing counterclockwise, when you look
down the x-axis.
xkzB ˆ
Problem 5.6 (a) A Phonograph record carries a uniform density of “static electricity” σ. If it rotates at angular velocity ω, what is the surface current density K at a distance r
from the center?
ω
z
r0
x
y
Problem 5.6(b) A uniformly charged solid sphere, of radius R and total charge Q, is centered at the origin and spinning at a
constant velocity ω about the z axis. Find the current density J at any point (r,θ,Φ)
within the sphere.
rθ
Φ
z
x
y
ω
P
Problem 5.5 A current I flows down a wire of radius a. (i) If it is uniformly distributed over
the surface,what is the current density K ?
(ii) If it is distributed in such a manner that the volume current density is inversely
proportional to the distance from the axis, what is J?
a
z
Problem: (a) A current I is uniformly distributed over a wire of circular cross-section, with radius a. Find the current density J. (b) Suppose the current density is proportional to the distance from the axis,J=ks. Find the total current in the wire.
a
z
The Continuity Equation
Q(t)The Arrows indicate charge leaving the volume V
tJ
…which is precisely the mathematical statement of local charge conservation.
• Why Steady Current is required here and which type of magnetic fields do steady currents give rise to??
• What is the form of the continuity equation in this case?
• ….and the “Biot-Savart Law”…
Magnetic Field of a Steady Current
The Biot-Savart Law:
The magnetic field of a steady current is given by:
/2
/\
4)( dl
r
rIrB
s
so
I
/dl
rs P
Problem: Find the magnetic field a distance
s from a long straight wire carrying a steady current I.
I
Ө1Ө2
Wire Segment
P
srsӨ
α
dL/ILong Wire
L/
Problem: Find the magnetic field a distance z above the center of a circular loop of
radius R, which carries a steady current I.
z
R'dl
rs
Problem:5.11 Find the magnetic field at point P on the axis of a tightly wound
solenoid (helical coil) consisting of n turns per unit length wrapped around a cylindrical
tube of radius a and carrying current I.
Ө1
Ө2
aP z
Problem: 5.9 Find the magnetic field at point P for each of the steady current
configurations shown below:
I
R ba
II
IP
R PI
I
Problem:5.45 A semicircular wire carries a steady current I. Find the magnetic field at a
point P on the other semicircle.
P
ӨR
I
Problem: 5.10(a) Find the force on the current carrying square loop due to a
current carrying wire
I
a
aI
s
Problem:5.46 The magnetic field on the axis of a circular current loop is far from uniform (it falls off sharply with increasing z). However,
one can produce a more nearly uniform field by
using two such loops a distance d apart.
dR
R
z=0
I
I
z
(a) Find the field B as a function of z, and show that ∂B/∂z is zero at the point midway between them (z=0). Now, if you pick d just right the second derivative of B will also vanish at the midpoint.
(b) Determine d such that ∂2B/∂z2=0 at the midpoint, and find the resulting magnetic field at the center.