1. Sources of Magnetic Fields

2. Besides magnetic poles, electric currents create magnetic fields.
There are two ways of calculating B produced by currents:
Biot-Savart Law: Field of a “current element”
(analogous to a point charge in electrostatics).
Ampère’s Law: An integral theorem similar to
Gauss’s law.

3. Biot-Savart Law dB r r θ I I ds (“permeability of vacuum”)
We get the total B by integrating ds along the wire:

4. Question: When is dB=0 ? How does dB vary with r ?
What is the direction of B at points a and b ? a
Current I in a straight wire b

5. Example 1 Find B at a distance R from a long straight wire y dB r f Rθ I x dx
(note sin θ = R/r = cos f)

6. Solution

7. Result: the magnitude of the field produced at a distance R from a long straight wire is:
The field lines form circles around the wire. Note the right-hand rule. I I

8. Quiz m0I/(2pa) m0I/(pa) zero none of the above P a I
What is the field at point P?
m0I/(4pa)
m0I/(2pa)
m0I/(pa)
zero
none of the above

9. Example 2 Circular loop in y-z plane. Find B. at the origin
at point (x, 0, 0) y I R x
(x, 0, 0) z

10. Solution y
At the Origin: dS R r z dB I

11. At the point (x,0,0): I ds dB • θ R r θ x θ r’ θ dB’ x I ds’
Pick a short segment ds where the wire crosses the y – axis:
I ds dB • θ R r θ x θ r’ θ
dB’ x
I ds’

12. Solution

13. current loop bar magnetI B
Field patterns
current loop B S N
bar magnet

14. Example 3 R2 R1 θo I O
Find the magnetic field at point O due to the current in the circuit. (Use the Biot-Savart Law on each segment)
b) Evaluate the expression for:
I = 20A, R1 = 10cm, R2 = 30cm, θo=120º

15. Solution