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.

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

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

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

Solution

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

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

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

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

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

Solution

Field patterns I B current loop B S N bar magnet

R2R2 R1R1 θoθo I I O 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 = 20A, R 1 = 10cm, R 2 = 30cm, θ o =120º Example 3