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 Unit vector:

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 A Long Straight Wire What is the meaning of “x”?

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

For Infinite Wire Semi-infinite Straight Wire 0 For Semi-Infinite Wire Even Function: Half the integral …

Off-axis for Long Straight Wire y x  Angle between 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.

Question

Step 1: Cut up the distribution into pieces Make use of symmetry! Need to consider only  B z 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 loop Magnetic Field of a Wire Loop

for z>>R: Magnetic Field of a Wire Loop Special case: far from the loop The magnetic field of a circular loop falls off like 1/z 3

For whole loop Magnetic Field of a Semicircle

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

What 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. X I I What is the net magnetic field at the center of the loop?

How does the magnetic field around a bar magnet look like? The Magnetic Field of a Bar Magnet NS

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:~ T Texas: ~2.5x10 -5 T Can use magnetic field of Earth as a reference to determine unknown field. Magnetic Field of Earth The magnetic field of the earth has a pattern that looks like that of a bar magnet

An electric dipole consists of two opposite charges – monopoles N S 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. What is the direction? S N What is the average current I? current=charge/second: One loop: The Atomic Structure of Magnets Electrons

Magnetic dipole moment of 1 atom: Method 1: use quantized angular momentum Orbital angular momentum: Quantum mechanics: L is quantized: If n=1: Magnetic Dipole Moment

Magnetic dipole moment of 1 atom: Method 2: estimate speed of electron Momentum principle: Circular motion:  – angular speed Magnetic Dipole Moment

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

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

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

Magnetic domains Hitting or heating while in a magnetic field can magnetize the iron Hitting or heating can also demagnetize Modern Theory of Magnets