Biot-Savart Law for a Single Charge

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 wireCurrent:1Four-step approach:Cut up the current distribution into pieces and draw B

Write an expression for B due to one piece

Add up the contributions of all the pieces

Check the resultMagnetic Field of Current Distributions2

Step 1:Cut up the current distribution into pieces and draw B.Origin: center of wireVector r:

Magnitude of r:

A Long Straight WireUnit vector:

3

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

:

B field due to one piece:

A Long Straight Wire

4

need to calculate only z component

A Long Straight Wire5

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

A Long Straight Wire6

Special case: xL

units:

A Long Straight Wire8

For Infinite Wire

Semi-infinite Straight Wire0For Semi-Infinite WireEven Function: Half the integral 9Half the integral for the infinite wire so half the result.Off-axis for Long Straight Wire

yxaAngle betweenSee Quest Course Resources for details (offaxisline.pdf)10

Right-hand Rule for WireConventional Current Direction11

QuestionCurrent carrying wires below lie in X-Y plane.12+zQuestion

13B

Step 1:Cut up the distribution into pieces

Make use of symmetry!

Need to consider only Bz due to one dlMagnetic Field of a Wire Loop14Note sense of dl-vector.Step 2: B due to one piece

Origin: center of loopVector r:

Magnitude of r:

Unit vector:

l:

Magnetic field due to one piece:Magnetic Field of a Wire Loop15Expression for delta-l is from Taylor expansion.Note that angle between dl and r is a right angle. Can prove by showing cross product is dl * r * sin(theta) = dl * r, so theta = 90.Step 2: B due to one piece

need only z component:

Magnetic Field of a Wire Loop

16Step 3: Sum the contributions of all pieces

Magnetic field of a loop along its axis:

Magnetic Field of a Wire Loop

17Step 4: Check the results

units:

direction:Magnetic Field of a Wire LoopCheck several pieces with the right hand ruleNote: Weve not calculated or shown the rest of the magnetic field18

Magnetic field of a loop:

Magnetic Field of a Wire Loop

19

Using general form (z=0) :

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

20

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/z321For whole loopMagnetic Field of a Semicircle

22What if we had a coil of wire?

For N turns:

single loop:A Coil of Wire23Make a usual coil and then with loops running in opposite directions.

far from coil:

far from dipole:

magnetic dipole moment: - vector in the direction of BMagnetic Dipole Moment24What is the magnetic dipole moment of a 3000-turn 35 cm rectangular coil that carries a current of 2 A?

for one turn

for N turns

Exercise25Equation is ~valid even if loop is not exactly circularThe 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 Dipole26What are the directions of the magnetic fields at the center of the loop?Exercise: a loop of radius R and a long straight wire. The center of the loop is 2R from the wire.XIIWhat is the net magnetic field at the center of the loop?27How does the magnetic field around a bar magnet look like?

The Magnetic Field of a Bar MagnetNS28Do experiment with compass and bar magnet to figure out.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 matterDifferences with E-field: B-field only interacts with some objects curly pattern only closed field linesMagnets and Matter29Superposition take 2 magnets and try to compensate the field detected by compass

Horizontal component of magnetic field depends on latitude

Maine:~1.5.10-5 TTexas: ~2.5x10-5 TCan 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 magnet30S-N are mixed up because of compass!Magnetic poles are 1300 km away from actual poles

Current is flowing to the right in a wire. The magnetic field at the position P pointsCurrent is flowing to the right in a wire. The magnetic field at the position P pointsChoice OneChoice TwoChoice ThreeChoice FourChoice FiveChoice SixA.B.

C.

D.What is the direction of the magnetic field inside the solenoid?What is the direction of the magnetic field inside the solenoid?Choice OneChoice TwoChoice ThreeChoice FourChoice FiveChoice SixA.B.C.D.

Current upward on side nearest youA 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)

Current clockwise; north pole on topCurrent counterclockwise, north pole on topCurrent clockwise; north pole on bottomCurrent counterclockwise, north pole on bottomB

C33An electric dipole consists of two opposite charges monopoles

NSBreak magnet:

S

NThere are no magnetic monopoles!Magnetic Monopoles34Do experiment: attach two magnets S_N-S_N act juts like one SN magnet. Break it got two magnets.The magnetic field of a current loop and the magnetic field of a bar magnet look the same.

What is the direction?SNWhat is the average current I?current=charge/second:

One loop:

The Atomic Structure of MagnetsElectrons35Also distnce dependence 1/r3.

Magnetic dipole moment of 1 atom:Method 1: use quantized angular momentumOrbital angular momentum:

Quantum mechanics: L is quantized:

If n=1:

Magnetic Dipole Moment36Also assume only 1 electron in one atom

Magnetic dipole moment of 1 atom:Method 2: estimate speed of electron

Momentum principle:

Circular motion:

angular speed

Magnetic Dipole Moment

37R = 0.5 x 10^-10 meters.Magnetic dipole moment of 1 atom:

Mass of a magnet: m~5gAssume magnet is made of iron: 1 mole 56 g6.1023 atomsnumber of atoms = 5g/56g . 6.1023 ~ 6.1022

Magnetic Dipole Moment381. Orbital motion

There is no motion, but a distributionSpherically symmetric cloud (s-orbital)has no Only non spherically symmetric orbitals (p, d, f) contribute to There is more than 1 electron in an atomModern Theory of Magnets392. Spin

Electron acts like spinning charge- contributes to Electron spin contribution to is of the same order as one due to orbital momentumNeutrons and proton in nucleus also have spin but their s are much smaller than for electron same angular momentum:

NMR, MRI use nuclear Modern Theory of Magnets40Alignment of atomic magnetic dipole moments:

most materialsferromagnetic materials:iron, cobalt, nickelModern Theory of MagnetsWhy are only some materials magnetics?41Magnetic domains

Hitting or heating while in a magnetic field can magnetize the ironHitting or heating can also demagnetizeModern Theory of Magnets42Alloys prevent iron from disordering after being magnetized.Hitting, heating above critical T destroys orderMagnetic domains

Why are there Multiple Domains?43

Multiplier effect:

Electromagnet:

Iron Inside a Coil44

Step 1: Cut up the distributioninto piecesBorigin: center of the solenoidStep 2: Contribution of one piece

one loop:Number of loops per meter: N/LNumber of loops in z: (N/L) zField due to z:

Magnetic Field of a Solenoid45

Step 3: Add up the contributionof all the piecesB

Magnetic field of a solenoid:Magnetic Field of a Solenoid46

Special case: R