Biot-Savart Law

Moving charge produces a curly magnetic field

B units: T (Tesla) = kg s-2A-1

Single Charge:

Biot-Savart Law

The Biot-Savart law for a short length of thin wire

Current:

Four-step approach:

1. Cut up the current distribution into pieces and draw B

2. Write an expression for B due to one piece

3. Add up the contributions of all the pieces

4. Check the result

Magnetic Field of Current Distributions

Step 1:Cut up the current distribution into pieces and draw B.

Origin: center of wire

Vector r:

Magnitude of r:

A Long Straight Wire

Step 2:Write an expression for B due to one piece.

Unit vector:

:

B field due to one piece:

A Long Straight Wire

need to calculate only z component

A Long Straight Wire

Step 3:Add up the contribution of all the pieces.

A Long Straight Wire

Special case: x<<L

A Long Straight Wire

What is the meaning of “x”?

Step 4: Check results

directionfar away: r>>L

units:

A Long Straight Wire

For Infinite Wire

Semi-infinite Straight Wire

0

∞

− ∞

− ∞

+∞

0

+∞

𝐵𝑠𝑒𝑚𝑖=𝜇0

4 𝜋𝐼𝑥

𝐵∞=𝜇0

4 𝜋2 𝐼𝑥

For Semi-Infinite Wire

Half the integral …

Right-hand Rule for Wire

Conventional Current Direction

QuestionCurrent carrying wires below lie in X-Y plane.

Question

𝐵𝑤𝑖𝑟𝑒=𝐵 h𝑒𝑎𝑟𝑡 tan (𝜃)¿ (2 ×1 0−5 T) tan (12° )T

Step 1:Cut up the distribution into pieces

Make use of symmetry!

Need to consider only Bz due to one dl

Magnetic Field of a Wire Loop

Step 2: B due to one piece

Origin: center of loop

Vector r:

Magnitude of r:

Unit vector:

l:

Magnetic field due to one piece:

Magnetic Field of a Wire Loop

Step 2: B due to one piece

need only z component:

Magnetic Field of a Wire Loop

Step 3: Sum the contributions of all pieces

Magnetic field of a loop along its axis:

Magnetic Field of a Wire Loop

Step 4: Check the results

units:

direction:

Magnetic Field of a Wire Loop

Check several pieces with the right hand rule

Note: We’ve not calculated or shown the “rest” of the magnetic field

Using general form (z=0) :

Special case: center of the loopMagnetic Field of a Wire Loop

for z>>R:

Magnetic Field of a Wire LoopSpecial case: far from the loop

The magnetic field of a circular loop falls off like 1/z3

For whole loop

Special case: at center of the semicircleMagnetic Field of a Semicircle

∫0

𝜋

¿ 12 ∫

0

2𝜋

❑

𝐵𝑧 , 𝑠𝑒𝑚𝑖=𝜇0

4𝜋𝜋 𝐼𝑅

𝐵𝑧 , ∆ 𝜃=𝜇0

4𝜋2𝜋 𝐼𝑅

∆ 𝜃2𝜋 What is for 1.5 loops?

What if we had a coil of wire?

For N turns:

single loop:

A Coil of Wire

far from coil: far from dipole:

magnetic dipole moment: - vector in the direction of B

Magnetic Dipole Moment

The magnetic dipole moment acts like a compass needle!

In the presence of external magnetic field a current-carrying loop rotates to align the magnetic dipole moment along the field B.

Twisting of a Magnetic Dipole

How does the magnetic field around a bar magnet look like?

The Magnetic Field of a Bar Magnet

N S

How do magnets interact with each other?Magnets interact with iron or steel, nickel, cobalt.

Does it interact with charged tape?

Does it work through matter?

Does superposition principle hold?Similarities with E-field:

• can repel or attract• superposition• works through matter

Differences with E-field:• B-field only interacts with some objects • curly pattern• only closed field lines

Magnets and Matter

Horizontal component of magnetic field depends on latitude

Maine: ~1.5.10-5 TTexas: ~2.5x10-5 T

Can use magnetic field of Earth as a reference to determine unknown field.

Magnetic Field of EarthThe magnetic field of the earth has a pattern that looks like that of a bar magnet

Current is flowing to the right in a wire. The magnetic field at the position P points

A. B.C. D.

What is the direction of the magnetic field inside the solenoid?

A. B. C. D.

Current upward on side nearest you

A current in the loop has created the magnetic field, B, shown below. What is the current direction in this loop if you look from the top? And which side of the loop is the north pole?(To get the pole, you need to replace the loop with a bar magnet that has the same field direction)

A. Current clockwise; north pole on top

B. Current counterclockwise, north pole on top

C. Current clockwise; north pole on bottom

D. Current counterclockwise, north pole on bottom

B

An electric dipole consists of two opposite charges – monopoles

NS

Break magnet:

S N

There are no magnetic monopoles!

Magnetic Monopoles

The magnetic field of a current loop and the magnetic field of a bar magnet look the same.

Batom 0

42z3 , R2I

What is the direction?

SNWhat is the average current I?

current=charge/second: I et

T 2 R

v RevI2

One loop:

eRvR

evR21

22

The Atomic Structure of Magnets

Electrons

eRv21

Magnetic dipole moment of 1 atom:

Method 1: use quantized angular momentum

Orbital angular momentum: RmvL

LmeRmv

meeRv

21

21

21

Quantum mechanics: L is quantized:

sJ , 341005.1nL

If n=1: 12

em

L 0.9 10 23 A m2 per atom

Magnetic Dipole Moment

eRv21

Magnetic dipole moment of 1 atom:

Method 2: estimate speed of electron

Momentum principle: netFdtpd

Circular motion:

drpdt

p vR

mv Fnet – angular speed

2

2

0

2

41

Re

Rmv

m/s 62

0

106.14

1

Rmev

1.3 10 23 A m2 /atom

Magnetic Dipole Moment

p p const

v / R

Magnetic dipole moment of 1 atom: /atommA 2 2310

Mass of a magnet: m~5g

Assume magnet is made of iron: 1 mole – 56 g

6.1023 atoms

number of atoms = 5g/56g . 6.1023 ~ 6.1022

magnet 6 1022 10 23 0.6 A m2

Magnetic Dipole Moment

1. Orbital motion

There is no ‘motion’, but a distribution

Spherically symmetric cloud (s-orbital)has no

Only non spherically symmetric orbitals (p, d, f) contribute to

There is more than 1 electron in an atom

Modern Theory of Magnets

Alignment of atomic dipole moments:

most materialsferromagnetic materials:iron, cobalt, nickel

Modern Theory of Magnets

2. Spin

Electron acts like spinning charge- contributes to

Electron spin contribution to is of the same order as one due to orbital momentum

Neutrons and proton in nucleus also have spin but their ‘s are much smaller than for electron

same angular momentum: me

21

NMR, MRI – use nuclear

Modern Theory of Magnets

Magnetic domains

Very pure iron – no residual magnetism spontaneously disordersHitting or heating can also demagnetize

Modern Theory of Magnets

Multiplier effect:

ironcoilnet BBB

coilnet BB

Electromagnet:

Iron Inside a Coil

Step 1: Cut up the distributioninto pieces

B

origin: center of the solenoid

Step 2: Contribution of one piece

Bz 0

42 R2I

R2 d z 2 3/2one loop:

Number of loops per meter: N/L

Number of loops in z: (N/L) z

Field due to z: Bz 0

42 R2I

R2 d z 2 3/2

NL

z

Magnetic Field of a Solenoid

Step 3: Add up the contributionof all the pieces

B

dBz 0

42 R2I

R2 d z 2 3/2

NL

dz

Bz 0

42 R2NI

Ldz

R2 d z 2 3/2 L /2

L /2

∫

Bz 0

42 NI

Ld L / 2

d L / 2 2 R2

d L / 2

d L / 2 2 R2

Magnetic field of a solenoid:

Magnetic Field of a Solenoid

Bz 0

42 NI

Ld L / 2

d L / 2 2 R2

d L / 2

d L / 2 2 R2

Special case: R<<L, center of the solenoid:

Bz 0

42 NI

LL / 2

L / 2 2

L / 2

L / 2 2

0

42 NI

L2

LNIBz

0 in the middle of a long solenoid

Magnetic Field of a Solenoid