Lots of details so far on how Magnetic Fields exert forces.But, where did those Magnetic Fields come from?

B-fields are created by moving charges (currents).

This seems quite opposite to what we might expect!

Sources of magnetic field

Magnetic Field created by a single moving charged particle at a position

20 ˆ

4 r

rvqB

×=v

v

πµ

Constant µ0 = 1.257 x 10-6 Tesla meter/Amp.

is a vector in the direction of , but with magnitude = 1 (often referred to as a unit vector). direction is from charge to where you are evaluating B !

Field created by a moving charge

rr

r̂r̂

rr

Field created by a moving chargeA new Right Hand Rule…

Thumb in direction of the motion, and fingers then curl in direction of the B-field.

20 ˆ

4 r

rvqB

×=v

v

πµ

20 ˆ

4 r

rvqB

×=v

v

πµClicker Question

What is the direction of the B-field at point P indicated?

A)Up and to the LeftB)Down and to the LeftC)Out of the PageD)Into the PageE)None of the Above

Point P

r

Example: Forces between two moving protons

zr

ev

r

rveB ˆ

4

ˆ

4 20

20

πµ

πµ =×=

v

v

yr

eFE ˆ

4

12

2

0πε=

r

( ) yr

vezx

r

evve

zr

evxve

BveFB

ˆ4

ˆˆ4

)(

ˆ4

)ˆ(

)(

2

220

20

20

πµ

πµ

πµ

=×−

=⎟⎠

⎞⎜⎝

⎛×−

=×−=r

r

r

Now the B-field created by many moving charges (i.e. current flowing in a wire).

I

20

20

20

ˆ

4

ˆ)(

4

ˆ

4

r

rldI

r

rvnqAdl

r

rvdQBd

d

d

×

=×

=×=

r

r

r

v

πµπ

µπ

µ

dl

Xr

dB

Biot-Savart first discovered this experimentally.

Magnetic field of a current element

I

20 ˆ

4 r

rldIBd

×=r

v

πµ

Magnetic field of a current element

20 ˆ

4 r

rLdIBd

×=v

v

πµ

∫∫×==2

0 ˆ

4 r

rLdIBdBtot

v

vv

πµ

This can be a very difficult integral to evaluate.

Magnetic field of a current-carrying wire

20 ˆ

4 r

rLdIBd

×=v

v

πµ

page) theinto( sin4 2

0 θπ

µr

dyIBd =v

222 xyr +=22//sin yxxrx +==θ

Now a little geometry

Magnetic field from an infinite straight wire

( ) 2/322

0

4 yx

xdyIBd

+=

πµv

∫∫+∞

∞− +==

2/3220

)(4 yx

xdyIBdB

πµvv

page) the(into 2

0

x

IB

πµ=

v

( ) 2/322

0

4 yx

xdyIBd

+=

πµv

Magnetic field from an infinite straight wire(cont.)

2

|| 0

R

IB

πµ=

v

This is a key result!

B-field a perpendicular distance x away from an infinite (or very long) wire.

Magnetic field from an infinite straight wire(cont.)

Clicker Question

i

A long wire has a current moving as shown.What is the direction of the B-field created by the wire just below the wire?

B=?

A) Into the PageB)Out of the PageC)To the rightD)DownE)None of the Above

20 ˆ

4 r

rLdiBd

×=v

v

πµ

L

r

Clicker Question

i

A long wire has a current moving as shown.What is the direction of the B-field created by the wire just above the wire?

B=?

A) Into the PageB)Out of the PageC)To the rightD)DownE)None of the Above

20 ˆ

4 r

rLdiBd

×=v

v

πµ

Lr

Power line has 500 Amps going through it.

What is the B-field strength 15 meters below on the ground?

Tm

AATm

R

IB 6

-60 107.6

)15(2

)500)(/10(1.26

2−×=×==

ππµ

Example with numbers

I I’ • Think of the Red Wire as creating a B-field.

• Then think of that B-field creating a force on the moving charges (current) in the Blue Wire.

Interaction between two current carrying wires

The two wires may exert forces on each other through Magnetic Interactions.

Interaction between two current carrying wires

BLIFvvv

×= '

rr

IB ˆ

20

πµ=

r

r

LIILBIF

πµ

2

'' 0==

Clicker Question

I IWhat is the direction of the Force

acting on the Blue Wire?

A)UpB)RightC)LeftD)Into the PageE)Out of the Page

BLIFvvv

×=

i1 i2

The B-field from the Red Wire at the location of the Blue Wire is into the page.

X

left)(2

102122 R

iLiBLiF

πµ=×=

vvv

Then the Force on the Blue Wire is to the left.

page) theinto(2

101 R

iB

πµ=

v

R

Clicker Question

i iWhat is the direction of the Force

acting on the Red Wire?

A)UpB)RightC)LeftD)Into the PageE)Out of the Page

BLIFvvv

×=

i1 i2

The B-field from the Blue Wire at the location of the Red Wire is out of the page.

.

right)(2

201211 R

iLiBLiF

πµ=×=

vvv

Then the Force on the Red Wire is to the right.

page) theofout (2

202 R

iB

πµ=

v

R

i i

Wires with parallel currents attract each other.

i

Wires with antiparallel currents repel each other.Try following the procedure we just outlined to confirm this for yourselves.i

What happens if we flip the direction of one current?

Interaction between current-carrying wires

What about the B-field at the center of a circular loop of wire of radius a and current I?

)(44 220

20

ax

dlI

r

dlIdB

+==

πµ

πµ

2/3220

)(4cos

ax

adlIdBdBx +

==π

µθ

Field through a circular loop

∫∫∫ +=

+== dl

ax

Ia

ax

adlIdBB xx 2/322

02/322

0

)(4

)(4 πµ

πµ

2/322

20

)(2 ax

IaBx +

= µ

Field through a circular loop

Which point A or B has the larger magnitude Magnetic Field?

A BC : The B-field is the same at A and B.

A

B

I I

Answer: Case B has the larger magnetic field. Use the Biot-Savart Law to get the directions of the B-field due to the two semi-circular portions of the loop. In A the two fields oppose each other; in B they add.

Clicker Question

20 ˆ

4 r

rldIBd

×=r

v

πµ

Clicker Question

2/322

20

)(2 ax

IaB

+= µIn the limit as x >> a, the expression

for the B-field becomes?

A)

B)

C)

3

20

2 x

IaB

µ=

a

IB

20µ=

)(2 22

20

ax

IaB

+= µ

a

I

x

B=?

xx

IaB ˆ

2 3

20µ=

v

xx

IAx

x

aIB ˆ

2ˆ

2 30

3

20

πµπ

πµ ==

a

I

x

B=?

* A = area of loop

AIv

v =µ Magnetic Dipole Moment

30

2 xB

µπ

µ v

v

=

Magnetic Dipole moment

AIv

v =µ

30

2 xB

µπ

µ v

v

=

Any current loop looks like a Magnetic Dipole far away.

The B-field drops as the distance^3 And depends on the Magnetic Dipole Moment.

Magnetic Dipole moment

A square loop of side length a of wire carrying current I is in a uniform magnetic field B. The loop is perpendicular to B (B out of the page). What is the magnitude of the net force on the wire?

I

B A: IaBB: 4IaBC: 2IaBD: 0E: None of these

Clicker Question

BLIFvvv

×=

Clicker Question

rB Bz= ⋅$ , where B = B(y) = A y

D

B

C

A

B stronger

B weakerE: net force is zero

The same loop is now in a non-uniform field.

where A is a constant. The direction of the net force is?

y

x

In a uniform B-field, regardless of the orientation between the B and the Magnetic Moment of the loop µ, the net force is always zero.

However, that does not mean the net torque is zero!

Uniform fields produce zero net force!!!

µ

If µ is not parallel to B, then there is a net torque.

∑ ×= Frv

vvτ

]sin)2/[(2)( θτ LBILBLIr∑ =××=vv

vv

θτ sin2BIL=v

BAIvv

v ×=τBv

vv ×= µτ

Magnetic torque

µ

Bv

vv ×= µτ Torque wants to twist the loop so that µ and B align.

The DC motor

Clicker Question

i1

i2

Two loops of wire have current going around in the same direction.

The forces between the loops is:

A)AttractiveB)RepulsiveC)Net force is zero.

Magnetic Field Lines and Flux

Gauss’s Law for Magnetic Fields

Magnetic monopoles (so far) do not exist!!!

There are no sources of magnetic flux

∫ =⋅ 0AdBrr

=>Flux through a closed surface:

Is there something like Gauss’ Law for Electric Fields that helps us solve for Magnetic Fields in a simpler way for cases with nice symmetries?

Ampere’s law

∫ =⋅loop thruIldB 0µ

vv

For any closed imaginary loop where the current is constant, the above relation is true.

Clicker Question

∫ =⋅loop thruIldB 0µ

vv

We need a sign convention for I(thru). Place the fingers of your right hand around the imaginary loop and your thumb points in the direction of positive I(thru).What is I(thru) in the below case where all three wires have 5 A?

A) I(thru) = +15 AmpsB) I(thru) = +5 AmpsC) I(thru) = -5 AmpsD) None of the above

XX

i

Place an imaginary circular loop of radius R around the wire.

∫ ⋅loop

ldBvv

Vector dot product!

Ampere’s law for a straight wire

Clicker Question

Look at the previous wire from the view where the wire current is coming out of the page.The Ampere circular loop is drawn in blue.

What is the relation of the vector dl that one integrates around the loop as shown and the Magnetic Field vector?A)ParallelB)Anti – Parallel C)PerpendicularD)Depends where in the loop for dl

i

dl

i

Thus, this is a special symmetry case where all around the loop the two vectors are parallel.

ldBvv

||In fact, also due to symmetry, we know thatthe magnitude of the B-field is also constant.

loop thearoundconstant || Bv

iaBldBldBlooploop

π2||||v

vv

vv

==⋅ ∫∫Where a is the radius of the loop.

∫ =⋅loop thruIldB 0µ

vv

IaB 02|| µπ =v

a

IB

πµ2

|| 0=v

Ampere’s law for a straight wire

b

r

J = constant

A long straight copper wire has radius b and carries a constant current of magnitude i. The current density of magnitude J=i/(πb2) is uniform throughout the wire. What is the current contained in the circular loop £, with radius r < b, centered on the wire's center as shown?

Clicker Question

A) B)

C) D) None of these.

b

ri

2

2

b

ri

3

3

b

ri

A)B ∝ r B) B = constantC) B ∝ 1/r D) B ∝ 1/r2

E) None of these

b

r

J = constant

A long straight copper wire has radius b and carries a constant current of magnitude i. The current density of magnitude J=i/(πb2) is uniform throughout the wire.How does the magnitude of the B-field a distance r < b from the center of the wire depend on r?

Clicker Question ∫ =⋅loop thruildB 0µ

vv

b

r

J = constant

∫ =⋅loop thruildB 0µ

vv

For r < b:

⎟⎟⎠

⎞⎜⎜⎝

⎛=

2

2

02||b

rirB µπ

v

20 2||

b

riB

πµ=

v

For r >= b:r

iB

πµ2

|| 0=v

|B|

rb

Clicker Question

We have determined that for the closed loop as drawn below the integral on the left is -10 Tesla meters. What is the direction of the current in the red wire shown below?

A) Into the PageB) Out of the PageC) LeftD) UpE) Down

∫ =⋅loop thruildB 0µ

vv

A single wire tightly coiled up into loops. Since it is a single wire, the current magnitude is the same in all parts of the coil.

Field of a solenoid

What is the B-field from a solenoid of N turns and length L?

Length = L

Number of Turns = N n = N/LCurrent Direction

Field of a solenoid

Side View

XXXXX

∫ =⋅loop thruIldB 0µ

vv

)(|| 0 nlIlBinside µ=v

l

Uniform B Inside

XXX

Zero B Outside

IL

NnIBinside 00|| µµ ==

v

B-Field Inside a Solenoid

# loops in length l

Field of a solenoid

Three long straight solenoids all of length L and all with the same (large) number of closely-packed turns N, all with the same current i, have different cross-sections as shown.

A

B

C

Which solenoid has the largest field |B| at its center?A) A B) B C) CD) All three have the same magnitude magnetic field

Clicker Question

Answers: The field is the same magnitude and uniform for all three solenoids. The field within a solenoid is B = µni. This depends only on the current i and the turns per length n. This formula does not depend on either the cross-sectional shape of the solenoid or the position within the solenoid.

Toroidal solenoid

Using Ampere’s Law: Ir

NBNIrB

πµµπ

22 00 =→=

If the “thickness” of the core is very small: nIBn

r

N02

µπ

=→≈

Projectile

Conducting Gas

I

X B

BLIFvvv

×=

How does a rail gun work?

Rail Guns are by far the most spectacular type of electromagnetic accelerators ever developed. They hold the record for fastest object accelerated of a significant mass, for the 16000m/s firing of a .1 gram object by Sandia National Research Laboratories' 6mm Hypervelocity Launcher, and they can also propel objects of very sizeable masses to equally impressive velocities, such as in the picture, where Maxwell Laboratories' 32 MegaJoule gun fires a 1.6 kilogram projectile at 3300m/s (that's 9 MegaJoules of kinetic energy!) at Green Farm research facility.

20 ˆ

4 r

rldIBd

×=r

v

πµ

20 ˆ

4 r

rvqB

×=v

v

πµ

Moving charges (currents) create B-fields.

Where are the moving charges?

Natural magnetism: Where is the B-field coming from?

Atoms have moving electric charges.

Natural magnetism: Where is the B-field coming from?

5Tesla 2

B 0 ≈=R

Iµv

Consider a single electron moving in a circular orbit around the nucleus.

Am

smC

πR

evI 4

10

619

106.7)10(2

)/103)(106.1(

2−

−

−

×=××==π

What is the B-field at the center (at the nucleus)?

Electron in orbit, but current in opposite direction.

Natural magnetism: Where is the B-field coming from?

In most materials all the electrons in orbit have random orientations.

Superposition of B-field vectors over many atoms gives B=0

Natural magnetism

In Ferromagnetic materials (Fe, Ni, Cr, some alloys containing these metals too), the atomic currents can all orient the same way, making a net B-field.

B

B

B

B

Ferromagnetism

Sometimes the material is fragmented into many domains (top) and is thus unmagnetized. Domains that are magnetized in the direction of the field grow larger.

Ferromagnetism

In all permanent magnets (just like a current loop)the B-field points away from one end (North)and towards the other end (South).

Very important convention.

Thus, these magnets act just like a current loop!i

Permanent magnets

What is the current direction in this loop? And which side of the loop is the north pole?

A. Current clockwise; north pole on topB. Current counterclockwise, north pole on topC.Current clockwise; north pole on bottomD.Current counterclockwise, north pole on bottom

i

Clicker Question

Atoms have a Magnetic Dipole Moments from the orbit of the electrons.

AIv

v =orbit)(µBut, amazingly electrons themselves also have a Magnetic Dipole Moment.

spin)or electron(µv

Electron “spin”

Is the electron really a fundamental point-like particle?Or does it have little charges moving around inside it?Or could the electron be a rotating sphere of charge?

We know from the world’s best microscopes (particle physics accelerators) that the electron is less than 10-18 meters across.

Thus, knowing the Magnetic Dipole Moment, we can calculate how fast the outside of the sphere must be rotating.

ondmeters/sec 105.2 14×=v

Electron “spin”

light) of (speed /103m/s105.2 814 smv ×>>×=

This velocity is faster than the speed of light, and thus not allowed in Einstein’s Theory of Relativity.

Thus, it may be that electrons just have an intrinsic property of Magnetic Dipole Moment.

This is not understood at any more fundamental level today.

Electron “spin”

It turns out in permanent magnets, the atoms have many electrons in orbits with different orientations.

Thus, it is often the intrinsic magnetic dipole moment of the electrons that is important and tends to be aligned into domains.

i

Clicker Question

A permanent bar magnet is broken in half. Do the pieces attract or repel?

1 2

A: AttractB: RepelC: Neither! There is no net force.

Answer: They attract. The bar magnet can be thought of as a coil of current. Parallel currents attract. The currents on theends are parallel so there is an attraction.

i i

Clicker Question

A permanent bar magnet is broken in half. The two pieces are interchanged, keeping their orientations fixed, as shown below. Do the pieces attract or repel?

1 2

2 1

A: AttractB: RepelC: Neither! There is no net force.

Answer: They attract. The bar magnet can be thought of as a coil of current. Parallel currents attract. When the magnet is broken and rearranged, the currents on the ends are parallel so there is an attraction.

Opposite Poles attract.

Similar Poles repel.

This is not new physics. It is identical to the physics of our current loops.

Now one can see why we can never break a magnet and get just one pole!