Sources of Magnetic Fields BiotSavart Law Besides magnetic

Sources of Magnetic Fields • Biot-Savart Law

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

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

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

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

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 P What is the field at point P? m 0 I/(4 pa) B) m 0 I/(2 pa) C) m 0 I/(pa) D) zero E) none of the above A) a I

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

Solution y At the Origin: d. S R z r d. B I

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

Solution

Field patterns I B current loop bar magnet S N B

Example 3 a) 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 = 20 A, R 1 = 10 cm, R 2 = 30 cm, θo=120º I I R 2 θo R 1 O