electrostatics in vacuum, in conductors and in the presence of linear dielectrics principle of...
TRANSCRIPT
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.