Sources of Magnetic Fields

1. Sources of Magnetic Fields • Biot-Savart Law

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 r dB r θ I I ds (“permeability of vacuum”) We getthe total B by integrating ds along the wire:

4. Quiz 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 R θ 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 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): 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. Field patterns I B B N S current loop bar magnet

14. Example 3 I I θo R2 R1 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