1. Electricity & Magnetism
Seb Oliver
Lecture 15:
Biot-Savart Law

5. Biot-Savart Law

6. 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

7. 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 …

8. 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

9. 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

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

11. 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

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

13. 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

14. Another Right-Hand Rule

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

16. Using the Biot-Savart Law
Example 1:
A wire loop

17. Examples of using the Biot-Savart Law
Some examples are quite hard and show how difficult it is to use this Law
In a real situation you might use a computer to calculate the magnetic field due to the wire
Next week we will discover Amperes Law which is often much easier to use

18. Magnetic Field from a Current Loopds
Direction: r
Magnitude:
B is always out of screen
ds is always  to r ds r dB

19. Using the Biot-Savart Law
Example 2:
A tight coil

20. Magnetic Field from a tight coil I.e. many current loopds
Direction: r
Magnitude:
as for a single loop
1 loopCurrent I
N loops Current  NI

21. Using the Biot-Savart Law
Example 3:
A long Straight wire

22. Magnetic Field from a Straight wireq ds x

23. Magnetic Field from a Straight wire
Magnitude:
Direction:
B is always out of screen
It turns out it is easier if we get everything in terms of q ds r dB

24. Magnetic Field from a Straight wire
Magnitude: …
Now we have to use calculus!

25. Other examples of Magnetic field
Centre of a wire loop radius R
Centre of a tight Wire Coil with N turns
Distance a from long straight wire

26. Magnetic Force Between Two Parallel Wires

27. Magnetic Force Between Two Parallel Wires
A current carrying wire in a magnetic field feels a magnetic force
A current carrying wire generates a magnetic field
Thus two current carrying wires will exert a force on each other

28. Magnetic Force Between Two Parallel Wires
Magnetic field at wire 1 from wire 2 I2 B2
Magnetic Force on wire 1 due to B2 (everything )

29. Magnetic Force Between Two Parallel Wires
Force on wire 1 due to wire 2
From Newton’s 3rd Law (& symmetry) Force on wire 2 due to wire 1
Force / unit length

30. Definition of Ampere
If the magnitude of the force per unit length between 2 parallel wires carrying identical currents and separated by 1m is 2×10-7 N/m then the current in each wire is 1A
Definition of a Coulomb: If a current of 1A is passing through a wire then 1C of charge passes a surface in 1s

31. Quiz
I1 = 2A, I2 = 6A
(a) F1 = 3F2
(b) F1 = F2
(c) F1 = F2/3

32. Summary Biot-Savart Law Force between two wires Definition of Ampere
(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