Lecture 17: THU 18 MAR 10 Ampere’s law Physics 2102 Jonathan Dowling André Marie Ampère (1775 – 1836)

Given an arbitrary closed surface, the electric flux through it is proportional to the charge enclosed by the surface. q Flux = 0! q Remember Gauss Law for E-Fields?

Gauss’s Law for B-Fields! No isolated magnetic poles! The magnetic flux through any closed “Gaussian surface” will be ZERO. This is one of the four “Maxwell’s equations”. There are no SINKS or SOURCES of B-Fields! What Goes IN Must Come OUT! There are no SINKS or SOURCES of B-Fields! What Goes IN Must Come OUT!

The circulation of B (the integral of B scalar ds) along an imaginary closed loop is proportional to the net amount of current traversing the loop. The circulation of B (the integral of B scalar ds) along an imaginary closed loop is proportional to the net amount of current traversing the loop. i1i1 i2i2 i3i3 ds i4i4 Thumb rule for sign; ignore i 4 If you have a lot of symmetry, knowing the circulation of B allows you to know B. Ampere’s law: Closed Loops

The circulation of B (the integral of B scalar ds) along an imaginary closed loop is proportional to the net amount of current traversing the loop. The circulation of B (the integral of B scalar ds) along an imaginary closed loop is proportional to the net amount of current traversing the loop. Thumb rule for sign; ignore i 3 If you have a lot of symmetry, knowing the circulation of B allows you to know B. Ampere’s law: Closed Loops

Sample Problem Two square conducting loops carry currents of 5.0 and 3.0 A as shown in Fig What’s the value of ∫B∙ds through each of the paths shown? Path 1: ∫B∙ds =   (–5.0A+3.0A) Path 2: ∫B∙ds =   (–5.0A–5.0A–3.0A)

Ampere’s Law: Example 1 Infinitely long straight wire with current i. Symmetry: magnetic field consists of circular loops centered around wire. So: choose a circular loop C so B is tangential to the loop everywhere! Angle between B and ds is 0. (Go around loop in same direction as B field lines!) R Much Easier Way to Get B-Field Around A Wire: No Calculus.

Ampere’s Law: Example 2 Infinitely long cylindrical wire of finite radius R carries a total current i with uniform current density Compute the magnetic field at a distance r from cylinder axis for: r < a (inside the wire) r > a (outside the wire) Infinitely long cylindrical wire of finite radius R carries a total current i with uniform current density Compute the magnetic field at a distance r from cylinder axis for: r < a (inside the wire) r > a (outside the wire) i Current out of page, circular field lines

Ampere’s Law: Example 2 (cont) Current out of page, field tangent to the closed amperian loop For r < R For r>R, i enc =i, so B=  0 i/2  R = LONG WIRE! For r>R, i enc =i, so B=  0 i/2  R = LONG WIRE! Need Current Density J!

R r B O Ampere’s Law: Example 2 (cont)

Solenoids The n = N/L is turns per unit length.

Magnetic Field in a Toroid “Doughnut” Solenoid Toriod Fusion Reactor: Power NYC For a Day on a Glass of H 2 O

Magnetic Field of a Magnetic Dipole All loops in the figure have radius r or 2r. Which of these arrangements produce the largest magnetic field at the point indicated? A circular loop or a coil currying electrical current is a magnetic dipole, with magnetic dipole moment of magnitude  =NiA. Since the coil curries a current, it produces a magnetic field, that can be calculated using Biot-Savart’s law:

Magnetic Dipole in a B-Field Force is zero in homogenous field Torque is maximum when  = 90° Torque is minimum when  = 0° or 180° Energy is maximum when  = 180° Energy is minimium when  = 0°

Neutron Star a Large Magnetic Dipole