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