Biot-Savart Law Moving charge produces a curly magnetic field Single Charge: The Biot-Savart law for a short length of thin wire Current: B units: T (Tesla) = kg s-2A-1 𝜇 0 4𝜋 = 10 −7 T∙m A

Magnetic Field of Current Distributions Four-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 result

A Long Straight Wire Step 1: Cut up the current distribution into pieces and draw B. Origin: center of wire Vector r: Magnitude of r: Unit vector:

A Long Straight Wire Step 2: Write an expression for B due to one piece. : 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 Has same distance dependence as E-field from long uniformly charged wire. What is the meaning of “x”?

A Long Straight Wire Step 4: Check results  direction  far away: r>>L  units:

Semi-infinite Straight Wire For Infinite Wire −∞ +∞ ∞ 𝐵 ∞ = 𝜇 0 4𝜋 2𝐼 𝑥 For Semi-Infinite Wire +∞ Half the integral for the infinite wire … so half the result. Even Function: Half the integral … 𝐵 𝑠𝑒𝑚𝑖 = 𝜇 0 4𝜋 𝐼 𝑥 −∞

Off-axis for Long Straight Wire y a Angle between ∆ 𝑙 𝑟 ∆ 𝐵 = 𝜇 0 4𝜋 1 𝑟 2 ∆𝑦 sin 𝛼 − 𝑧 Rewrite ∆𝑦 in terms of ∆𝛼 x See Quest Course Resources for details (offaxisline.pdf)

Right-hand Rule for Wire Conventional Current Direction

Question Current carrying wires below lie in X-Y plane. +z

Question 𝐵 𝑤𝑖𝑟𝑒 = 𝐵 𝑒𝑎𝑟𝑡ℎ tan⁡(𝜃) =(2×1 0 −5 T) tan⁡(12°) B 𝐵 𝑤𝑖𝑟𝑒 = 𝐵 𝑒𝑎𝑟𝑡ℎ tan⁡(𝜃) =(2×1 0 −5 T) tan⁡(12°) =4.3×1 0 −6 T

Magnetic Field of a Wire Loop Step 1: Cut up the distribution into pieces Note sense of dl-vector. 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: Expression 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. Magnetic field due to one piece: Unit vector: l:

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: Check several pieces with the right hand rule Note: We’ve not calculated or shown the “rest” of the magnetic field

Magnetic Field of a Wire Loop Magnetic field of a loop:

Magnetic Field of a Wire Loop Special case: center of the loop Using general form (z=0) :

Magnetic Field of a Wire Loop Special case: far from the loop for z>>R: The magnetic field of a circular loop falls off like 1/z3

Magnetic Field of a Semicircle Special case: only 𝐵 𝑧 at center of the semicircle For whole loop 0 𝜋 = 1 2 0 2𝜋 𝐵 𝑧,𝑠𝑒𝑚𝑖 = 𝜇 0 4𝜋 𝜋𝐼 𝑅 𝐵 𝑧,∆𝜃 = 𝜇 0 4𝜋 2𝜋𝐼 𝑅 ∆𝜃 2𝜋 What is ∆𝜃for 1.5 loops?

A Coil of Wire single loop: What if we had a coil of wire? For N turns: Make a usual coil – and then with loops running in opposite directions.

Magnetic Dipole Moment far from coil: far from dipole: magnetic dipole moment:  - vector in the direction of B

Exercise What is the magnetic dipole moment  of a 3000-turn 35 cm rectangular coil that carries a current of 2 A? for one turn for N turns Equation is ~valid even if loop is not exactly circular

Twisting of a Magnetic Dipole 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.

Exercise: a loop of radius R and a long straight wire Exercise: a loop of radius R and a long straight wire. The center of the loop is 2R from the wire. I 𝐵 𝑤𝑖𝑟𝑒 = 𝜇 0 4𝜋 2𝐼 𝑟 𝐵 𝑙𝑜𝑜𝑝 = 𝜇 0 4𝜋 2𝜋𝐼 𝑅 X I What are the directions of the magnetic fields at the center of the loop? What is the net magnetic field at the center of the loop? 𝐵 𝑙𝑜𝑜𝑝 − 𝐵 𝑤𝑖𝑟𝑒 = 𝜇 0 4𝜋 2𝜋𝐼 𝑅 − 𝜇 0 4𝜋 2𝐼 2𝑅 = 𝜇 0 𝐼 4𝜋𝑅 2𝜋−1

The Magnetic Field of a Bar Magnet How does the magnetic field around a bar magnet look like? N S Do experiment with compass and bar magnet to figure out.

Magnets and Matter 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 Superposition – take 2 magnets and try to compensate the field detected by compass Differences with E-field: B-field only interacts with some objects curly pattern only closed field lines

Magnetic Field of Earth The magnetic field of the earth has a pattern that looks like that of a bar magnet Horizontal component of magnetic field depends on latitude Maine: ~1.5.10-5 T Texas: ~2.5x10-5 T S-N are mixed up because of compass! Magnetic poles are 1300 km away from actual poles Can use magnetic field of Earth as a reference to determine unknown field.

Current is flowing to the right in a wire Current is flowing to the right in a wire. The magnetic field at the position P points Current is flowing to the right in a wire. The magnetic field at the position P points A. B. C. D. Choice Five Choice Six Choice Four Choice Three Choice Two Choice One

What is the direction of the magnetic field inside the solenoid? Current upward on side nearest you A. B. D. C. Choice Five Choice Six Choice Four Choice Three Choice Two Choice One

Current clockwise; north pole on top 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) Current clockwise; north pole on top Current counterclockwise, north pole on top Current clockwise; north pole on bottom Current counterclockwise, north pole on bottom B C

Magnetic Monopoles An electric dipole consists of two opposite charges – monopoles Break magnet: N S S N Do experiment: attach two magnets S_N-S_N – act juts like one SN magnet. Break it – got two magnets. There are no magnetic monopoles!

The Atomic Structure of Magnets The magnetic field of a current loop and the magnetic field of a bar magnet look the same. Electrons What is the direction? One loop: What is the average current I? S N current=charge/second: Also – distnce dependence 1/r3.

Magnetic Dipole Moment Magnetic dipole moment of 1 atom: Method 1: use quantized angular momentum Orbital angular momentum: Quantum mechanics: L is quantized: Also assume only 1 electron in one atom If n=1:

Magnetic Dipole Moment Magnetic dipole moment of 1 atom: Method 2: estimate speed of electron Momentum principle: Circular motion: – angular speed R = 0.5 x 10^-10 meters.

Magnetic Dipole Moment Magnetic dipole moment of 1 atom: 6.1023 atoms Mass of a magnet: m~5g Assume magnet is made of iron: 1 mole – 56 g number of atoms = 5g/56g . 6.1023 ~ 6.1022

Modern Theory of Magnets 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 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: NMR, MRI – use nuclear 

Modern Theory of Magnets Why are only some materials magnetics? Alignment of atomic magnetic dipole moments: ferromagnetic materials: iron, cobalt, nickel most materials

Modern Theory of Magnets Magnetic domains Alloys – prevent iron from disordering after being magnetized. Hitting, heating above critical T – destroys order Hitting or heating while in a magnetic field can magnetize the iron Hitting or heating can also demagnetize

Why are there Multiple Domains? Magnetic domains

Iron Inside a Coil Multiplier effect: Electromagnet:

Magnetic Field of a Solenoid Step 1: Cut up the distribution into pieces Step 2: Contribution of one piece origin: center of the solenoid one loop: B Number of loops per meter: N/L Number of loops in z: (N/L) z Field due to z:

Magnetic Field of a Solenoid Step 3: Add up the contribution of all the pieces B Magnetic field of a solenoid:

Magnetic Field of a Solenoid Special case: R<<L, center of the solenoid: in the middle of a long solenoid