Straight line currents

IldB 0

Straight line currents

Φ1

Φ2

Loop

Wire

…we have a bundle of straight wires and some of the wires passes

through the loop…

I1

I2I3

I4

I5

Loop

For surface current and volume currents:

adr

rrKrB

s

s 2

0 ˆ)(

4)(

d

r

rrJrB

s

s2

0 ˆ)(

4)(

…for any arbitrary current distribution

The Divergence and Curl of B

rs P(x,y,z)

dτ/(x/,y/,z/)

Applications of Ampere’s Law:

…in

• Straight wire

• Sheet of Current

• Long Solenoid

• Toriodal Coil

…Straight Wire

Amperian Loop

I

s

B

…Infinite Sheet of Current

z/\

xKK

K

yz

xy

xAmperian Loop

Long Solenoid

K

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.

Ia

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.

a bb

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.

a

R

b

Iin

x

y

z

…find the magnetic field B at a point (a) on the x axis at x=2R and (b) on the y axis at

y=2R.

a

R

b

Iin

x

y

z

…(c) at a point inside the hole.

a

R

b

Iin

x

y

z

Problem: 5.14 A thick slab extending from z=-a to z=+a carries a uniform volume

current ,

xJJ ˆ

Find the magnetic field, as a function of z, both inside and outside the slab.

x

z

y+a-a

Problem:5.16 A large parallel plate capacitor with uniform surface charge density σ on the upper plate and –σ on the lower is moving with a constant speed v, as shown below.

+σ

-σ

v

v

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.

+σ

-σ

v

v

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.

Problem: …the magnetic flux through the end face of a

solenoid…

AK

Comparison of Magnetostatics and Electrostatics

The divergence and curl of the electrostatic field are:

,0

,0

E

E

…together with the Boundary conditions determine the field uniquely.

Comparison of Magnetostatics and Electrostatics

The divergence and curl of the Magnetostatics field are:

,

,0

0 JB

B

…together with the Boundary conditions determine the field uniquely.

Magnetic Vector Potential

,0

B

…permits us to introduce a Vector Potential A in Magnetostatics:

AB

Problem: A spherical shell, of radius R, carrying a uniform surface charge σ, is set

spinning at angular velocity ω. Find the vector potential A it produces at point P.

ωr

z

R

x

yΦ/

Ө/Ψ

rs

r/ da/

P

ω

Rr

Pz

ydRrrR

RrA ˆ

cos2

sincos

2

sin)(

22

30

…Expressions for the Magnetic Field Inside & Outside the Spherical Shell

are:

)(

,ˆ3

2

ˆsinˆcos3

2

0

0

Rr

zR

rR

AB

(Inside the Spherical Shell)

…Expressions for the Magnetic Field Inside & Outside the Spherical Shell

are:

)(

ˆsinˆcos23 3

40

Rr

rr

R

AB

(Outside the Spherical Shell)

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 σ and spinning at an angular

velocity ω).

…A Spinning Shell…

)(

ˆsinˆcos23

&

),(

ˆsinˆcos3

2

3

40

0

Rr

rr

R

Rr

rR

AB

ω

K

K

R

(r=R at the surface)

Problem: Find the vector potential of an infinite solenoid with n turns per unit

length, radius R, and current I.

Problem:5.22 Find the magnetic vector potential of a finite segment of straight

wire, carrying a current I.

zrs

sI

z1

z2

o

Problem: 5.24 If B is uniform, show that,

BrrA

2

1)(

works.

Is this result unique, or there are other functions with same divergence and curl.

y

x

A

Ax

Ay

zzBB ˆ0

r

o

Here, ‘r ’ is the projection of ‘vector-r’ on x-y plane.

Problem: 5.23 What current density would produce the vector potential,

kA

(where k is a constant),

in cylindrical coordinates ?

,

ˆˆ)(1

s

k

sB

Now

zs

kzsk

ss

AB

Multipole Expansion of the Vector Potential

I

dr/=dl/r/

rrs

P

O

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.

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.

z

x

y

I

w

The Magnetic Field of a Pure Dipole

x

Ө

Φy

z

m

r P

ˆsinˆcos24

,ˆ

4)(

30

20

rr

mAB

r

rmrAdipole

Problem:5.34 Show that the magnetic field of a dipole can be written in

coordinate free form as:

mrrmr

rBdip

ˆˆ.31

4)(

30

Ө

Φy

z

m

r P

x

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.

R

z

(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 z>>R.

Problem:5.35 A phonograph record of radius R, carrying a uniform surface charge σ, is rotating at

constant angular velocity ω. Find its dipole moment.

ω

z

R0

y

x

Fields due to “Pure” dipole & Physical dipole.

Problem: 5.55 A Magnetic dipole zmm ˆ0

is situated at the origin, in an otherwise

uniform magnetic field

.ˆ0 zBB

Show that there exists a spherical surface, centered at the origin,

through which no Magnetic field line passes.

Magnetostatic Boundary Conditions

K

Babove┴

Bbelow┴

Magnetostatic Boundary Conditions

K BIIabove

BIIbelow

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.

These Results can be Summarized as:

)ˆ(0 nKBB belowabove

where, ‘n’ cap is the

unit Vector perpendicular to the

Surface pointing upwards.

Problem : Show that the Vector Potential A is continuous, but its derivative inherits a discontinuity

across any boundary.

belowabove AA

Kn

A

n

A belowabove0

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.

R zs

s

zA

sAsss

A

z

As

z

AA

sA szsz ˆ)(

1ˆˆ1

J

END OF THE MAGNETOSTATICS