Chapter 30 Ampere’s Law

Chapter 30 Ampere’s Law PHYS 2326-19

Concepts to Know • Magnetic Forces between wires • Ampere’s law • Solenoid • Toroid

Magnetic Forces Between Wires • Chapter 30.2 • Similar to forces between two charged particles • Give 2 parallel wires 1 & 2 separated by distance a with currents I1 & I2 What are the forces between them? B2 1 I1 F1 2 I2 a

Magnetic Force Between 2 Wires • Wire 2 generates a field B2 • For a long straight wire B2 is from eqn 30.5 • Force F1 on wire 1 comes from eqn 29.10 • Using magnitudes eqn 30.12 Using right hand rules for directions B2 is out of plane and F1 is down towards wire 2

Ampere’s Law • The line integral of B·ds (B dot ds) is equal to μo I (the permeability of free space times the current I)

The Toroid • A ring or torus wrapped with a wire • Create amperian loop – dashed line • by symmetry (assume wire uniform all the way around) – the field is constant B and tangent to it so B·ds = B ds • Wire passes through N times (4 shown)

The Toroid • Since there are N turns, total current through the loop is NI • Apply Ampere’s law

. . . . . . . x x x x x x The Solenoid • Ideal solenoid is when length >> radius • B is uniform and parallel inside ideal solenoid • Consider loop 2 a rectangle w*l in area • Side 3 is far away so B = 0, side 2&4 B is perpendicular with the sides • Side 1 is parallel and in a uniform field B so that the magnitude is Bl B loop 2 2 3 l 1 4 w loop 1

Example 1 • Given a rectangular loop of 10cm x 20cm (a by b) with counter clockwise current I=2A what is the magnitude and direction of magnetic field at the center point P • B direction right hand rule = out of page • B = Bab+Bbc+Bcd+Bda = 2Bab + 2Bbc a d I P b c

Example 1 From eqn 30.4 for the field produced by a long wire. The book shows sines which is opposite/adjacent • B = 8.78E-6 T

Example 2 • Long straight vertical wire carries 10A A rectangular coil of wire located near it carries 5 A, a = 0.10m, b=0.30m, c=0.50m. Force on loop? Force on top = -Force on bottom b c I2 I1 a x

Example 2

Example 3 Use Ampere’s Law to determine magnetic field Choose a circle centered around the wire so B is tangent at every point and uniform in distance from the wire B dl r I

Example 3 • Dot product B·l = Bdl • Same magnitude at each point (symmetry) • B is constant so

Example 4Inside and outside a Toroid • Find the magnetic field a) inside and b) outside a tightly wound toroid of N turns

Chapter 30 Ampere’s Law