Sources of Magnetic Fields

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.

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

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

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)

Solution

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

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

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

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

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’

Solution

current loop bar magnet I B Field patterns current loop B S N bar magnet

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

Solution