Tài liệu Chapter X Magnetic Field doc

By analogy to electric interaction we introduce for magnetic interaction the concept of magnetic field which is the means of transfering magnetic interactions: A magnet sets up a magneti

GENERAL PHYSICS II

Electromagnetism

&

Thermal Physics

Chapter X Magnetic Field

§1 Magnetic interaction and magnetic field

§2 Magnetic forces on a moving charged particle

and on a current-carrying conductor

§3 Magnetic field of a current – magnetic field calculations

§4 Amper’s law and application

§1 Magnetic interaction and magnetic field

1.1 Magnetic phenomena:

 Some history:

 Magnetic effects from natural magnets have been known for a longtime Recorded observations from the Greeks more than 2500 yearsago

 The word magnetism comes from the Greek word for a certain type ofstone (lodestone) containing iron oxide found in Magnesia, a district innorthern Greece

 Properties of lodestones: could exert forces on similar stones and

could impart this property (magnetize) to a piece of iron it touched

 Bar magnet: a bar-shaped permanent magnet It has two poles: N and S

Like poles repel; Unlike poles attract

We say that the magnets can interact each with other This kind of

 We have known that the means of transfering interactions between

electric charges is electric field By analogy to electric interaction we

introduce for magnetic interaction the concept of magnetic field which

is the means of transfering magnetic interactions:

A magnet sets up a magnetic field in the space around it and the

second magnet responds to that field

 The direction of the magnetic field at any point is defined as the

direction of the force that the field would exert on a magnetic north pole

of compass needle

The earth itself is a magnet

Note that for the earth magnet:

the geographical pole ≠the magnetic pole N

S

North geographic poleSouth magnetic

pole

South geographic pole

North magnetic pole

N

S

1.2 Magnetic field vector and magnetic field lines:

By analogy to electric field vector E we can introduce magnetic

field vector B :

+ The direction of magnetic field vector at each point in the space

can be defined experimentally by a compass

+ The mathematical expression for magnetic field vector (magnitudeand direction) will be defined below (the law of Biot and Savart)

Magnetic field lines can be drawn in the same manner as electric

field lines (direction and density)

N S

Magnetic Field Lines of

a bar magnet

Electric Field Lines

of an Electric Dipole

N S

Magnetic Monopoles ?

 Perhaps there exist magnetic charges, just like electric charges Such anentity would be called a magnetic monopole (having + or - magnetic

charge)

 How can you isolate this magnetic charge?

Try cutting a bar magnet in half:

• Many searches for magnetic monopoles no monopoles have

ever been found !

N S

Even an individualelectron has a

magnetic “dipole”!

Source of Magnetic Fields?

 What is the source of magnetic fields, if not magnetic charge?

 Answer: electric charge in motion!

 e.g., current in wire surrounding cylinder (solenoid) produces verysimilar field to that of bar magnet

 Therefore, understanding source of field generated by bar magnet

lies in understanding currents at atomic level within bulk matter

Motions of electrons on orbits and intrinsic motions producemagnetic field

Orbits of electrons about nuclei

Intrinsic “spin” of electrons (more important effect)

• The force F on a charge q moving with velocity v through a region

of space with magnetic field B is given by:

§2 Magnetic forces on a moving charged particle

and on a current-carrying conductor:

2.1 Magnetic force on a moving charge:

B v

q

F   





Magnetic Force: (Lorentz force)

• In the formula B is measured in Tesla (T): 1T = 1 N / A.m

Example 1:

 Two protons each move at speed v (as

shown in the diagram) in a region ofspace which contains a constant B field

in the -z-direction Ignore the interaction

between the two protons

 What is the relation between themagnitudes of the forces on the twoprotons?

(a) decreases (b) increases (c) stays the same

C – Inside the B field, the speed of each proton:

 Two independent protons each move

at speed v (as shown in the diagram)

in a region of space which contains a

constant B field in the -z-direction

Ignore the interaction between the two

protons

 What is the relation between the

magnitudes of the forces on the twoprotons?

F B

v q

 Two independent protons each move

at speed v (as shown in the diagram)

in a region of space which contains a

constant B field in the -z-direction.

Ignore the interaction between the two

protons

 What is F 2x, the x-component of

the force on the second proton?

(a) F2x < 0 (b) F2x = 0 (c) F2x > 0

B

• To determine the direction of the force, we use the corkscrew rule

(or right-hand rule)

• As shown in the diagram, F2x < 0

B v

v

v

(a) decreases (b) increases (c) stays the same

v v

 Two protons each move at speed v (as

shown in the diagram) in a region of

space which contains a constant B

field in the -z-direction Ignore the

interaction between the two protons

 Inside the B field, the speed ofeach proton:

Although the proton does experience a force (which deflects it), this forcealways ⊥to v Therefore, there is no possibility to do work, so kinetic

energy is constant and | v | is constant

1) Turn on electron ‘gun’

qV

mv 2 2

1

qB

mv

R 

2) Turn on magnetic field B

3) Rearrange in terms of measured values, V, R and B

2

B R

V

m q 

m

q V

• The scheme for the “gun”

• The principle of a selector:

2.2 Magnetic force on a current-carrying conductor:

 Consider a current-carrying wire in the

presence of a magnetic field B.

 There will be a force on each of the charges

moving in the wire What will be the total force

dF on a length dl of the wire?

 Suppose current is made up of n charges/volume

each carrying charge q and moving with velocity v

through a wire of cross-section A.

 d F Id l B





B v

q dl nA F

d   



nAvq dt

q dt nAv dt

dq

I   ( ) 

B v

The case for a straight length of wire L carrying

a current I, the force on it is: F I L B

2.3 Magnetic force and torque on a current loop:

 Consider loop in magnetic field as on

right: If field is to plane of loop, the

net force on loop is 0!

• If plane of loop is not to field, there will

be a non-zero torque on the loop!

 Force on top path cancels force on

bottom path (F = IBL)

 Force on right path cancels force

on left path (F = IBL)

• What is the force on section a-b of the loop?

a) zero b) out of the page c) into the page

• What is the force on section b-c of the loop?

a) zero b) out of the page c) into the page

4) What is the net force on the loop?

a) zero b) out of the page c) into the page

A square loop of wire is carrying

current in the counterclockwise

direction There is a horizontal uniform

magnetic field pointing to the right

ab: Fab = 0 = Fcd since the wire is parallel to B.

bc: Fbc = ILB RHR: I is up, B is to the right, so F points into the screen

Calculation of Torque:

• Suppose the loop has width w (the side

we see) and length L (into the screen)

The torque is given by:

 Note: if loop B, sin= 0  = 0

maximum occurs when loop parallel to B

F r

• We can define the magnetic dipole moment of a current loop as

follows:

+ direction: to plane of the loop in

the direction the thumb of right hand

points if fingers curl in the direction of

current

+ magnitude:   AI

• Torque on loop can then be rewritten as:

• Note: if loop consists of N turns, = NAI

Concerning to magnetic dipole

moment we have an analogue:

(N loops  N bar magnets)

2.4 Potential energy of a magnetic dipole:

 Work must be done to change the

orientation of a dipole (current loop) in

the presence of a magnetic field

• Define a potential energy U (with zero at

position of max torque) corresponding to

this work





θ τ dθ U

90  



θ μB θdθ U

2.4 Potential energy of a magnetic dipole:

2.5 Applications: 1) Galvanometers ( ≡Dial Meters)

We have seen that a magnet can exert a torque on a loop of current– aligns the loop’s “dipole moment” with the field

– In this picture the loop (and hence theneedle) wants to rotate clockwise

– The spring produces a torque in theopposite direction

– The needle will sit at its equilibriumposition

Current increased

 μ= I • Area increases

 Torque from B increases

 Angle of needle increases

Current decreased

 μdecreases

 Torque from B decreases

 Angle of needle decreases

2) Motors:

Turn the current on and slightly tip the loop

Restoring force from the magnetic torque

Oscillations

Now turn the current off, just as the loop’s μis aligned with B

Loop “coasts” around until its μis ~antialigned with B

Turn current back on

Magnetic torque gives another kick to the loop

Continuous rotation in steady state

B

Free rotation

of spindle

Even better

 Have the current change directions every half rotation

 Torque acts the entire time

Two ways to change current in loop:

1 Use a fixed voltage, but change the circuit (e.g., break

connection every half cycle

 Direct-current (DC) motors

2 Keep the current fixed, oscillate the source voltage

 Alternative-current (AC) motors

VS I

t

2.6 The Hall Effect

 Which charges carry current?

 Positive charges moving

counterclockwise experience

upward force

 Upper plate at higher potential

 Negative charges movingclockwise experience upwardforce

 Upper plate at lower potentialEquilibrium between electrostatic & magnetic forces:

 This type of experiment led to the discovery (E Hall, 1879) that

current in conductors is carried by negative charges

 Soon after the discovery of the Hall effect, it was obseverd that

some material, particularly some semiconductors have Hall emf

opposite to that of the metals, as if their charge carriers were

positive charged (the holes in some semiconductors).

Summary on magnetic forces

 Force due to B on a moving charged particle F = q v x B

Remark on Electric & Magnetic Dipoles Analogy

F r

τ      τ    r  F 

(per turn)

E q

§3 Magnetic field of a current – magnetic field calculations:

 Now we study how can calculate the magnitude and the direction

of magnetic field produced by a current

 In order to do that let us recall two ways to calculate electric field:

q S

Analogously we have for magnetic field:

• Two ways to calculate

0

2

ˆ 4

μI dl r dB

d B

     0

I

N 10

• Calculate field at point P using

dx π

I

μ dB

θ

x

R θ

d π

I

μ B

0

sin 4

θ πR

dθ π

I

μ B

π

sin 4

0

0





b) Calculation of the B-field due to a current-carying circular loop

x

•

z

R R

 Circular loop of radius R carries

current I Calculate B along the axis of

2.3 Force between two parallel current-carrying wires:

• We know that a current-carrying wire can experience force from aB-field

• We know that a a current-carrying wire produces a B-field

• Therefore: We expect one current-carrying wire to exert a force onanother current-carrying wire:

• Current goes together  wires come together

• Current goes opposite  wires go opposite

• Calculate force on length L of wire b due to

field of wire a:

The field at b due to a is given by:

 Calculate force on length L of wire a due to

L I

a b

§4 Amper’s law and application:

I l

d B

     0

I

Line integral of magnetic field

vector around a closed path

Current “enclosed” by that path

4.1 Formulation of Amper’s law:

For magnetic field there exists a law which plays the same role as

the Gauss’s law for electric field

The Amper’s law:

• Calculate field at distance R

from wire using Ampere's Law :

dl



R

I

 Choose loop to be circle of radius R centered

on the wire in a plane to wire

2

0



 B   d l   B ( π 2 R )

 Line integral around the closed path:

 Current enclosed by path = I

I μ πRB 0

 Apply Ampere’s Law:

Example: B-field of a straight wire:

4.2 Application of Amper’s law:

a) B field inside a long wire:

 Suppose a total current I flows through

the wire of radius a into the screen as

shown

 Calculate B field as a function of r, the

distance from the center of the wire

• Inside the wire: (r < a)

• Outside the wire: ( r > a )

a







b) B Field of an infinite current sheet

 Consider an infinite sheet of current described

by n wires/length each carrying current i into

the screen as shown Calculate the B field

• What is the direction of the field?

• Symmetry  vertical direction

• Calculate using Ampere's law for a square of

x

x x

x x

x x

w

c) B Field of a Solenoid:

 A constant magnetic field can (in principle) be produced by an

infinite sheet of current In practice, however, a constant magneticfield is often produced by a solenoid

• If a << L, the B field is to first order contained within the

solenoid, in the axial direction, and of constant magnitude In

• A solenoid is defined by a current i flowing

through a wire that is wrapped n turns per unit

length on a cylinder of radius a and length L

L

a

 To correctly calculate the B-field, we should use

Biot-Savart, and add up the field from the different loops

 To calculate the B field of the solenoid using Ampere's Law, we

need to justify the claim that the B field is nearly 0 outside the

solenoid (for an infinite solenoid the B field is exactly 0 outside)

• The fields are in the same direction in the region

between the sheets (inside the solenoid) and

cancel outside the sheets (outside the solenoid)

• To do this, view the infinite solenoid from the

side as 2 infinite current sheets

x x x x x x x x x x x

• • • • • • • • • • • • • •

• Draw square path of side w:

Bw l

B 

Note:

Use the wrap rule to find the B-field: wrap your fingers in the

direction of the current, the B field points in the direction of the thumb(to the left)

Since the field lines leave the left end of solenoid, the left end is the

north pole

The magnetic field of a solenoid is essentially identical to that of a barmagnet.

The big difference is that we can turn the solenoid on and off ! It

attracts/repels other permanent magnets; it attracts ferromagnets,etc

b) B field of a toroid:

• Toroid defined by N total turns

with current i

• B=0 outside toroid! (Consider

integrating B on circle outside toroid )

 To find B inside, consider circle of radius r,

centered at the center of the toroid

x x

x

x

x x

(2 r

B l

Example B-field Calculations:

 Inside a Long Straight Wire

 Infinite Current Sheet

B  0

πr

Ni

μ B