Biot-Savart Law for a Single Charge Electric field of a point charge: Moving charge makes a curly magnetic field: B units: T (tesla) = kg s -2 A -1 The Biot-Savart law for a short length of thin wire conventional current: in the direction of positive charges  0 permeability of free space

Calculate B-field strength produced by I=100A current through a straight wire at a radius r=0.1m from the wire. B=10 -7 *2*100/0.1=2*10 -4 TB earth = 5*10 -4 T

Clicker Current carrying wires below lie in X-Y plane.

Right-hand Rule for Wire Conventional Current Direction

Clicker

Electric fields: produced by charges Magnetic fields: produced by moving charges charged tape Any magnetic field? Frame of Reference

Must use the velocities of the charges as you observe them in your reference frame! There is a deep connection between electric field and magnetic fields (Einstein’s special theory of relativity) Frame of Reference

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 =

What if we had a coil of wire? For N turns: single loop: A Coil of Wire

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

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

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 Exercise

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 NS

Horizontal component of magnetic field depends on latitude Maine:~ T Indiana: ~ T Florida: ~ 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

Similarities with E-field: both attraction and repulsion can be observed the superposition principle is valid the interaction passes through matter Differences with E-field: while charged objects interact with all other types of objects, magnets (B-field) interacts only with some objects magnets are permanently magnetic, while regular objects can acquire or lose their charge two negatively charged objects always repel each other, whereas two magnets may repel or attract depending on their orientation Magnets and Matter

The Atomic Structure of Magnets A single atom can be a magnetic dipole: An electron orbiting around nucleus An electron spinning on its own axis Rotational motion in the nucleus (protons and neutrons have spin about their own axes and can also orbit around inside the nucleus). S N Simple Bohr model

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

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 (can be ignored in modeling bar magnet) same angular momentum: NMR, MRI – use nuclear  Modern Theory of Magnets

Magnetic domains Very pure iron – no residual magnetism spontaneously disorders Heating can demagnetize Modern Theory of Magnets These which are aligned with B tend to grow

Multiplier effect: Electromagnet: Iron Inside a Coil

Magnetic domains Why are there Multiple Domains?