Magnetostatics - Universitetet i oslofolk.uio.no/ravi/cutn/elec_mag/8_magstatics2.pdf · iron possess a property we call magnetism that exertsiron, possess a property we call magnetism

Magnetostatics

P.Ravindran, PHY041: Electricity & Magnetism 22 January 2013: Magntostatics

Magnetic Fields

• We saw last lecture that some substances, particularly iron possess a property we call magnetism that exertsiron, possess a property we call magnetism that exerts forces on other magnetic materialsW l th t i l ti h ( ti• We also saw that single magnetic charges (magnetic monopoles) did not existW h i fi ld h b i fili• We saw that magnetic fields, shown up by iron filings look similar to electric dipole fields

• Also that magnetic fields seem to be associated with moving charges

• What is this "magnetic force"? How is it related to and distinguished from the "electric" force?


Magnetic Forces

• Consider a positive charge qmoving in the field of a B

g

magnet with velocity , experimentally we find:

vp y

1. If q moves in the +z direction and the field points in the +yand the field points in the +ydirection then the force is in the –x direction The force is

F

the –x direction. The force is proportional to the velocity and the fieldand the field

2. If q moves in the +x direction h f h dthe force is in the +z direction, again proportional to andB

v

Magnetic Forces

3. If qmoves in the +y direction h fthere is no force

4. If q is at rest there is no force

5. The force is proportional to B

6. The force is proportional to the sign and magnitude of q

The magnetic force on a moving charge is proportional to q, vp and B, where vp is the velocity component

F

q, p , p y pperpendicular to the field, while the direction of is perpendicular to both and and depends on the sign of

F

B

v


q F qv B

Lorentz Force

• We can add the effect of an Electric Field and get the “Lorentz Force”the Lorentz Force

• The force F on a charge q moving with velocity vg q g ythrough a region of space with electric field E and magnetic field B is given by:g g y

BvqEqF x x x x x x

B

B

B

x x x x x xx x x x x x

v v

v

Fx x x x x x

q qF = 0

qF

Reminder: The Cross Product

• The cross (vector) product of two vectors is a third vector – Remember the dot (scalar) product multiplied two vectors to produceRemember the dot (scalar) product multiplied two vectors to produce

a scalar

BBAA X BB = CC

AA

• The magnitude of CC is given by:C = AB sin

CCC AB sin

• The direction of CC is perpendicular to the plane• The direction of CC is perpendicular to the plane defined by AA and BB, and in the direction defined by the right hand rule, rotating from A to B.


by the right hand rule, rotating from A to B.

UIUC

Reminder: The Cross Product

• Cartesian components of the cross product:

CC = AA X BBC C A A X BBCX = AY BZ ‐ BY AZ

CY = AZ BX ‐ BZ AX

BB

CZ = AX BY ‐ BX AYAANote: B X A = - A X BCC Note: B X A A X B

•Drawing 3-dimensional vectors, conventionally

– a vector going into the slide

– a vector coming out of the slide

Right Hand RuleRight Hand Rule


Motion in a magnetic field F qv B

Three points are arranged in a uniform magnetic field. The B field points into the xg pscreen. Consider the force on a positively charged particle in the following conditions

x

1) It is located at point A and is stationary.

•v=0 The magnetic force is zero

yz

g

2) The positive charge moves from point A toward B.

• in direction B in RH r le sa s F in•v in x direction, B in z, RH rule says F in –y

•The direction of the magnetic force on the particle is to the left

3) The positive charge moves from point A toward C.

•Rotate our x axis to be along the direction A‐C


• F will be perpendicular to that line and upwards

Motion of a Charge in a Magnetic FieldThe ’s represent field lines pointing into the page. A positively charged particle of massThe s represent field lines pointing into the page. A positively charged particle of mass m and charge q is shot to the right with speed v. By the right hand rule the magnetic force on it is up. Since v is to B, F = FB = q v B. Because F is to v, it has no tangential component; it is entirely B q , g p ; ycentripetal. Thus F causes a centripetal acceleration. As the particle turns so do v and F, and if B is uniform the particle moves in a circle. This is the basic idea behind a particle accelerator like Fermilab. Since F is a centripetal force, F = FC = m v2 / R. Let’s see how Cspeed, mass, charge, field strength, and radius of curvature are related:

R

FB = FC F

q v B = m v2 / R

v

+q,

Bmv


m R = q B

Question 1y

B

y

1

2

v

v

• Two protons each move at speed v in the x‐y plane (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 thexz

2 v– What is the relation between the magnitudes of the forces on the two protons?

( ) F F (b) F F ( ) F F(a) F1 < F2 (b) F1 = F2 (c) F1 > F2

•The magnetic force is given by:•The magnetic force is given by: sinF qv B F qvB

• In both cases the angle between v and B is 90°

• Therefore F1 = F2.

Question 2y

• Two protons each move at speed v in the x‐y plane (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. B

y

1

2

v

v– What is F2x ,the x‐component of the force on the second proton? xz

2 v

(a) F2x < 0 (b) F2x = 0 (c) F2x > 0(a) F2x < 0 (b) F2x 0 ( ) 2x

•To determine the direction of the force we use theTo determine the direction of the force, we use theright-hand rule.

BvqF

•The directions of the forces are shown in the diagram

F 0

BvqF


F2x < 0

Question 3y

• Two protons each move at speed v in the x‐y plane (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. B

y

1

2

v

v– Inside the B field, the speed of each proton: xz

2 v

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

•Although the proton does experience a force (which deflects•Although the proton does experience a force (which deflects it), this is always to . Therefore, there is no possibility to do work

v

W F l Fl

• So kinetic energy is constant and is constant vcosW F l Fl


Trajectory in a Constant B Field• Suppose charge q enters B-field with velocity v as shown below. What will

be the path q follows?

x x x x x xx x x x x xx x x x x xx x x x x x

v Bx x x x x xx x x x x x x x x x x xqx x x x x x

FFv

RR• Force is always to velocity and B.

– Path will be circle. F will be the centripetal force needed to keep the


p pcharge in its circular orbit, radius R.

Radius of Circular Orbit

• Lorentz force:

qvBF x x x x x xx x x x x xqvBF

• centripetal acc:

2

x x x x x xx x x x x x

v Bx x x x x xx x x x x x

Rv

a2

x x x x x x

F qFv

R

x x x x x x

R• Newton's 2nd Law:

2

maF Rv

mqvB2

qBmv

R This is an important result, with useful experimental


qB consequences !

Ratio of charge to mass for an electronRatio of charge to mass for an electron

• In 1897 J J Thomson used• In 1897 J.J.Thomson used a cathode ray tube to measure e/m for an /electron

•Used an electric and magnetic field in opposition to cancel force and thus deflection of the electron

•Electron accelerated through a voltage V by the “electron gun” giving it kinetic energy 21

2m v e V

•Velocity when it enters the fields2

2eVvm


e/m for an electron•In an electric field alone the spot is deflectedspot is deflected

•Then apply a magnetic field at right angles (Force in oppositeright angles (Force in opposite direction) until the deflection is reduced to zeroreduced to zero•Force due to electric field ˆ

EF e E k

ˆF B B k

•Force due to magnetic field

•When the fields cancelBF ev B evB k

ˆ ˆ E BEF F qEk qvBk vB

When the fields cancel E B B

2eV2e E

1 11 .7 5 8 8 2 0 1 0e Ck gm


vm

22m VB

Measurement of particle energies

•Many experiments using particles measure their velocity (energy or y ( gymomentum) by measuring their curvature in a magnetic field

•Cloud chambers

•Bubble chambersBubble chambers

•Magnetic spectrometers


Magnetic dipole moment

• A current loop behaves like a little bar magnet aligning with a magnetic field.g g g

• The magnitude of the dipole moment is NiA

• The direction of the dipole moment vector is NiA

pgiven by right hand rule


i

Biot Savart LawBiot‐Savart Law

• Moving charges are affected by magnetic fields; similarly moving charges (currents) create magnetic fields.


The Biot‐Savart law

Magnetic field intensity B at position P by a conductor with current I

0 RdIdB

dI dB

30

4 RRdIdB

2/104 7 AN

P(x,y,z)Q(x’,y’,z’)

I'rrR

dB0 /104 7 AN

: permeability of free space

c R

RdIB 30

4

I

integrating

OdRJB v

0


In 3‐D, dv

RB

v

v 3

0

4

Ampere’s LawAmpere s Law

• Ampere’s law is to magnetism what Gauss’s law is to electrostatics.

IldB encIldB 0. • This method works in cases with high

t h th ti f th B fi ldsymmetry where the properties of the B field can be inferred, figured out, whatever.


Ampere’s circuit lawAmpere s circuit law

Amper’s lawAmper s law

The line integral of B around a closed path is the same as the net current Ienc enclosed by the path.

II

d

enccIdB 0

C

Bd

C

sdrJldrBs

vc

)()( 0

Integral form of Ampere’s Law

sc


g p

Ampere’s Law: differential form

Ampere’s Law IdB

p

Ampere’s Law c

IdB 0

sdrBldrBsc

)()(

S

CsdrJIs

v

)(00

)()( 0 rJrB v Ampere’s Law: differential form )()( 0p

3rd Maxwell equation


Magnetic flux

• magnetic flux

s

dsB Unit: Wb

• Gauss’s Law for magnetostatic field

0s dsBIntegral form

0 BDifferntial form

4th Maxwell eq

Thers is no magnetic charge-monopole..

4th Maxwell eq.


g g p

Current enclosedCurrent enclosed

• Current density J is the current per unit area through a wire. g

AdJIenc . dJenc .


Faraday’s law and Lenz’s lawFaraday s law and Lenz s law

d ’ l d ib h i fi ld• Faraday’s law describes how magnetic fields can create voltages i.e. we are now connecting magnetic and electric phenomena

• In words a time varying magnetic flux through y g g ga circuit loop creates a voltage difference between the ends of the loop.p

• Lenz’s law indicates the polarity of the voltage


Faraday’s lawFaraday s law• V is the induced voltage between the ends• Vind is the induced voltage between the ends of the loop, M is the magnetic flux through the loop

dtdV M

ind

• Flux through 1 loop is given bydt

Arealoop

M AdB1

.

• Flux through N loops is N MArealoop1


Faraday’s law: big pictureFaraday s law: big picture

• The fundamental point of Faraday’s law is that a time-varying magnetic flux, M, leads to an y g g , M,induced voltage and thus an E-field.

• In the briefest of terms• In the briefest of terms– A changing magnetic field produces an electric field.


Lenz’s law

• The voltage induced is always such as to keep the flux through the circuit constant. The gdirection of the voltage is oriented to create a current in the loop such that the flux remainscurrent in the loop such that the flux remains the same.


Magnetic Force on a Current Carrying WireA section of wire carrying current to the right is shown in a uniform magnetic field. We can imagine positive charges moving to right, each feeling a magnetic force out of the page. This will cause the wire to bow outwards. Shown on the right is the view as seen when l ki t th N l f b Th d t t iflooking at the N pole from above. The dots represent a uniform

mag. field coming out of the page. The mag. force on the

Swire is proportional to the field strength, the current, and the length of the wire. S

I . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

NI. . . . . . . . . . . . . . . . . . . . . . . .

BP.Ravindran, PHY041: Electricity & Magnetism 22 January 2013: Magntostatics

N B

Continued…

Magnetic Force on a Wire (cont.)

Current is the flow of positive charge. As a certain amount of charge, q, moves with speed v through a wire of length L, the

F = qv B

g , q, p g g ,force of this quantity of charge is:

F = qv BOver the time period t required for the charge to traverse the length of the wire we have:

F = (q / t)v t B. . . . . . . . . . . . . . . . . . . . . . . .

length of the wire, we have:

I

. . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . Since q / t = I and v t = L, we can write:

F = ILB. . . . . . . . . . . .

Bwrite:


where L is a vector of magnitude L pointing in the direction of I.

Electric Motor I

I

I

} dF

I

} d

I B

Current along with a magnetic field can produce torque. This is the basic idea behind an electric motor. Above is a wire loop (purple) carrying a current provided by some power source like a battery. The current loop is submerged in an external field. From F = ILB, the force vectors in black are perpendicular to their wire segments. The net force on the loop is zero, but the net torque about the center is nonzero. The forces on the left and right wires produce

i h i f h ( h i i h h ) hno torque since the moment arm is zero for each (they point right at the center). However, the force F on the top wire (in the background) has a moment arm d, so it produces a torque Fd. The bottom wire (in the foreground) produces the same torque. These torques work t th t t t th l ti l t i l i t h i l


together to rotate the loop, converting electrical energy into mechanical energy.

Continued…

Electric Motor (cont.)As the loop turns it eventually reaches a vertical position (the plane of the loop parallel to the field). This is when the moment arms of the forces on the top and bottom wires are the longest so this isthe forces on the top and bottom wires are the longest, so this is where the torque is at a max. 90° later the loop will be perpendicular to the field. Here all moment arms and all torques are zero. This is the equilibrium point. The angular momentum of the loop, however, will allow it to swing right through this position.

Now is when the current must change direction, otherwise the torques will attempt to bring the loop back to the equilibrium. This would amount to simple harmonic motion of the loop which is notwould amount to simple harmonic motion of the loop, which is not particularly useful. If the current changes direction every time the loop reach equilibrium, the loop will spin around in the same p q , p pdirection indefinitely. Although a battery only pumps current in one direction, the change in direction of current can be accomplished ith h l f t t


with help of a commutator.

Electromagnets: Straight WirePermanent magnets aren’t the only things that produce magnetic fields. Moving chargesPermanent magnets aren t the only things that produce magnetic fields. Moving charges themselves produce magnetic fields. We just saw that a current carrying wire feels a force when inside an external magnetic field. It also produces its own magnetic field. A long straight wire produces circular field lines centered on the wire. To find the direction of the g pfield, we use another right hand rule: point your thumb in the direction of the current; the way your fingers of your right hand wrap is the direction of the magnetic field. Bdiminishes with distance from the wire. The pics at the right show cross sections of a current carrying wire.

II fI out of page, B counterclockwise

I into page, B clockwise

B


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Magnetostatics - Universitetet i oslofolk.uio.no/ravi/cutn/elec_mag/8_magstatics2.pdf · iron possess a property we call magnetism that exertsiron, possess a property we call magnetism