Electricity & Magnetism

Seb OliverLecture 14: Biot-Savart Law

Summary: Lecture 13

• Practical uses of moving charge in magnetic field

• Lorentz Force

• Force on Wire BLF IB

EBvF qq qB

mvr

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

r

ds

dB

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

• 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

r

ds

dB

Biot – Savart Law

• All these results could be summarised by one “Law”

2

ˆ

r

rdsB

Id

20 ˆ

4 r

rdsB

Id

Putting in the constant

Where 0 is the permeablity of free spaceA

Tm104 7

0

Magnetic Field from Biot-Savart Law

• B = dB1+dB2+…+dBi

• I.e. B =dB

20 ˆ

4 r

rdsB

Id

r1

ds1

dB1

r2

ds2

dsi

dBi

ri

dB2

2

0 ˆ

4i

iiIr

rdsB

•We can use the Biot-Savart law to calculate the magnetic field due to any current carrying wire

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

20 ˆ

4 r

rdsB

Id

r1

ds1

dB1 dB1

r1

ds

r

dB

Another Right-Hand Rule

Magnetic Field from Biot-Savart Law

20 ˆ

4 r

rdsB

Id

dB1

r1

c.f.

20 ||4

1||

rE

Q

Of course there is no such thing as an isolated current segment!

Summary

• Biot-Savart Law – (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

a

II

l

F

2

210

a

IB

2

0

R

IB

20

R

NIB

20

20 ˆ

4 r

rdsB

Id