Electricity & Magnetism
Seb Oliver
Lecture 14:
Biot-Savart Law
Summary: Lecture 13
• Practical uses of moving charge in
magnetic field
F  qv  B  qE
• Lorentz Force
• Force on Wire
FB  I L  B
mv
r
qB
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
• 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

Biot & Savart’s
Results
ds
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 
Biot – Savart Law
• All these results could be summarised by
one “Law”
ds  rˆ
dB  I
r
Putting in the constant
2
 0  ds  rˆ
dB    I
2
 4  r
Where 0 is the permeablity of free space
 0  4 10
7
Tm
A
Magnetic Field from
Biot-Savart Law
 0  ds  rˆ
dB    I
2
 4  r
•We can use the Biot-Savart law
to calculate the magnetic field
due to any current carrying wire dB1
• B = dB1+dB2+…+dBi
• I.e. B =SdB
ds i  rˆi
 0 
B   I 
2
4

 
ri
r1
dB2 dB
i
r2
ri
ds2
ds1
dsi
One Example of using the BiotSavart 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
r1
ds1
r
ds
dB1
r1
dB
 0  ds  rˆ
dB    I
2
 4  r
Another Right-Hand Rule
Magnetic Field from Biot-Savart
dB
Law
1
r1
c.f.
 0  ds  rˆ
dB    I
2
 4  r
1
Q
| E |
40 | r |2
Of course there is no such thing as an isolated current segment!
Summary
• Biot-Savart Law
–
–
–
–
 0  ds  rˆ
dB    I
2
 4  r
0 I
(Field produced by wires)
B
2R
Centre of a wire loop radius R
 NI
B 0
Centre of a tight Wire Coil with N turns
2R
Distance a from long straight wire
• Force between two wires
• Definition of Ampere
0 I
B
2a
F  0 I1 I 2

l
2a