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??

Current carrying wires

v

B (Due to wire 1)

1 2

F X

)(

BvQFM

In the presence of an Electrostatic Field E

)(

BvEQF

…justified by the experiments as well…

Problem of a charged particle in a Cyclotron:

z

x

yv

F

X BR

Q

…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

Practice Problem:

m

QB

m

qEa

where

ttta

x

tta

y

m

,

,

,cossin2

,sin2

2

z

x

y

Emcosωt

B

Answer

Magnetic Forces do not Work

0)(

dtvBvQldFdW magmag

Homework Problem: Example 5.3

….the Types of Current distributions

• Line Currents

• Surface Currents

• Volume Currents

• Line Currents

v

vI

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

y

z

x

I a

1

2

3

4

dlK

Flow

• Surface Currents

dl

IdK

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

• Volume Currents

Jda

Flow

da

IdJ

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.

This arrangement is known as a Helmholtz coil; it’s a convenient

way of producing relatively uniform fields in the laboratory.

dR

R

z=0

I

I

z