o Aim of the lecture  Calculation of Magnetic Fields Biot-Savart Law Magnetic field, B  Ampres Law B field Case of static E field o Main learning outcomes  familiarity with Calculation of Magnetic Fields from currents Biot-Savart Law Amperes Law Lecture 8

o We understand: o the source of electric field is the electric charges, o But there are no magnetic monopoles, o which means no magnetic charges o So where do magnetic fields come from? The answer is that it must be remembered There is a VERY close connection between electric and magnetic fields A magnetic field is just an electric field in motion or An electric field is just a magnetic field in motion ( loosely speaking – there is just one field, it is an electromagnetic field It becomes pure magnetic or electric when the charge is stationary)

o Hence it is possible to have a magnetic field when there are no magnetic charges. o What is needed is electric charges in motion  This is a CURRENT However: o The shape of the field that can be made is not the same A stationary electric charge makes a monopole electric field Two opposite electric charges makes an electric dipole field A string of electric charges moving in a loop (a current loop) makes a dipole magnetic field There is no monopole magnetic field observed in nature simply because there are no magnetic monopoles seen

Monopole electric field (electric charge) (remember this cannot be exactly correct in a 2-D drawing) Dip0le electric field (electric charges) Dipole magnetic field (electric charges) Same general field pattern using a single current loop DIPOLE

More detail on how to compute Magnetic Fields caused by currents s ds o The Biot-Savart Law  tells for each current element, I ds  the size and direction of the B field produced o The total field is then the sum of all such elements.

ds This current element Produces a B field element of constant magnitude on this ring Also a constant magnitude on this ring The element contributes everywhere in space according to the Biot-Savart Formula

ds To get the total field here Need to add all the contributions from all the current elements Vector Integral needed ∫ B= ∫ Note: in principle o All currents in universe o But because of 1/r 2 o Only nearby matter

Biot Savart

What this says is simply that integrating the B field round any closed loop will give you an answer which is proportional to the current flowing through the loop. It doesn't matter what shape the loop is (this is the case where the electric field is constant – see next slide)

Ampère The complete version is this:

This equation relates the magnetic field to the motion of charges and the change in electric field. In many cases the electric field is constant, so that the second term can be dropped. This is the case for fixed current distributions, such as currents in wires. ( which we did on the previous transparency ) J is current density

So this is Ampere's Law in differential form It is telling us how the magnetic and electric fields are related to the electric charges. (You are not expected to manipulate Maxwell’s equations in this course)

Next Lecture: Faraday’s Law of Induction