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# Straight line currents - PowerPoint PPT Presentation

Straight line currents. Loop. Wire. Φ 2. Φ 1. Straight line currents. …we have a bundle of straight wires and some of the wires passes through the loop…. I 5. I 3. I 2. I 1. Loop. I 4. For surface current and volume currents:. r s. P(x,y,z). d τ / (x / ,y / ,z / ).

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Loop

Wire

Φ2

Φ1

Straight line currents

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

I5

I3

I2

I1

Loop

I4

rs

P(x,y,z)

dτ/(x/,y/,z/)

…for any arbitrary current distribution

The Divergence and Curl of B

Applications of Ampere’s Law:

…in

• Straight wire
• Sheet of Current
• Long Solenoid
• Toriodal Coil
…Straight Wire

Amperian Loop

s

I

B

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.

I

a

a

b

Problem: 5.15 Two long coaxial solenoids each carry current I, but in opposite directions as shown. The inner solenoid (radius a ) has n1turns per unit length, and the outer one(radius b)has n2turns per unit length.

b

y

R

Iin

a

x

b

z

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.

y

R

Iin

a

x

b

z

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

z

+a

y

-a

x

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

Find the magnetic field, as a function of z,

both inside and outside the slab.

v

v

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.

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.

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.

Comparison of Magnetostatics and Electrostatics

The divergence and curl of the

electrostatic field are:

…together with the Boundary conditions

determine the field uniquely.

Comparison of Magnetostatics and Electrostatics

The divergence and curl of the Magnetostatics field are:

…together with the Boundary conditions determine the field uniquely.

Magnetic Vector Potential

…permits us to introduce a Vector Potential

A in Magnetostatics:

P

z

ω

z

P

rs

r

ω

r

Ψ

Ө/

R

da/

r/

R

y

Φ/

x

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.

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

(Inside the Spherical Shell)

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

(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).

ω

R

K

K

…A Spinning Shell…

(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.

z2

z1

o

I

z

s

rs

Problem: 5.24 If B is uniform, show that,

works.

Is this result unique, or there are other

functions with same divergence and curl.

Ay

A

y

Ax

r

x

o

z

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

(where k is a constant),

in cylindrical coordinates ?

Problem: 5.60(a) Work out the Multipole expansion for the vector potential for a volume currentJ.(b)Write down the Monopole potential and prove that it vanishes.(c)Write the corresponding dipole momentm.

z

I

w

y

x

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

P

r

Ө

m

y

Φ

Problem:5.34 Show that the magnetic field of a dipole can be written in coordinate free form as:

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.

z

R

(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 radiusR, carrying a uniform surface charge σ, is rotating at constant angular velocity ω. Find its dipole moment.

z

ω

0

R

y

x

Problem: 5.55 A Magnetic dipole

is situated at the origin, in an otherwise

uniform magnetic field

Show that there exists a spherical surface,

centered at the origin,

through which no Magnetic field line

passes.

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:

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.

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.

s

R

z

s

J