Electricity & Magnetism Seb Oliver Lecture 14: Biot-Savart Law

Summary: Lecture 13 Practical uses of moving charge in magnetic field Lorentz Force Force on Wire

Biot-Savart Law

Introduction We have discussed how an existing magnetic field influences moving charges (and thus currents) We have not yet discussed the origin of magnetic fields We will now see that currents (moving charges) produce magnetic fields This can be thought of as the basic mechanism by which all magnetic fields are produced

History 1819 Hans Christian Oersted discovered that a compass needle was deflected by a current carrying wire Then in 1920s Jean-Baptiste Biot and Felix Savart performed experiements to determine the force exerted on a compass by a current carrying wire There results were as follows …

Jean-Baptiste Biot & Felix Savart’s Results dB the magnetic field produced by a small section of wire ds a vector the length of the small section of wire in the direction of the current r the positional vector from the section of wire to where the magnetic field is measured I the current in the wire  angle between ds & r dB r  ds

Biot & Savart’s Results dB perpendicular to ds dB perpendicular to r |dB| inversely proportional to |r|2 |dB| proportional to current I |dB| proportional to |ds| |dB| proportional to sin q

Biot – Savart Law All these results could be summarised by one “Law” Putting in the constant Where m0 is the permeablity of free space

Magnetic Field from Biot-Savart Law We can use the Biot-Savart law to calculate the magnetic field due to any current carrying wire dB2 dB1 dBi dsi r2 ri B = dB1+dB2+…+dBi I.e. B =SdB r1 ds2 ds1

One Example of using the Biot-Savart Law Direction of the field around a long wire

Magnetic Field from Biot-Savart Law We can use the Biot-Savart law to see the direction of the field due to a wire segment dB1 dB1 r1 r1 ds1 ds r dB

Another Right-Hand Rule

Magnetic Field from Biot-Savart Law dB1 r1 c.f. Of course there is no such thing as an isolated current segment!

Summary: Lectures 14 & 15 Biot-Savart Law Force between two wires (Field produced by wires) Centre of a wire loop radius R Centre of a tight Wire Coil with N turns Distance a from long straight wire Force between two wires Definition of Ampere