Problem 4 A metal wire of mass m can slide without friction on two parallel, horizontal, conducting rails. The rails are connected by a generator which delivers a constant current i to the circuit. There is a constant, vertical magnetic field, perpendicular to the plane of the rails. If the wire is initially at rest, find its velocity as a function of time. i B l generator

The angular velocity

Uniform magnetic field,

Uniform, When a charged particle has velocity components both perpendicular and parallel to a uniform magnetic field, the particle moves in a helical path. The magnetic field does no work on the particle, so its speed and kinetic energy remain constant.

Example: A proton ( ) is placed in the uniform magnetic field directed along the x-axis with magnitude T. Only the magnetic force acts on the proton. At t=0 the proton has velocity components Find the radius of the helical path, the angular speed of the proton, and the pitch of the helix (the distance traveled along the helix axis per revolution).

Current carrying wires 1820 Hans Christian Oersted Hans Christian Ørsted

Ampere’s Law The field produced by an infinite wire

Problem 6 An infinitely long, hollow cylindrical wire has inner radius a and outer radius b. A current i is uniformly distributed over its cross-section. Find the magnetic field everywhere.

Biot-Savart Law Infinitesimally small element of a current carrying wire produces an infinitesimally small magnetic field (Also called Ampere’s principle) is called permeability of free space

Problem 2 R 2R

Field of a Current Carrying Loop R r x (along x)

Problem 1 Consider two infinitely long, parallel wires a distance d apart. Find the force between them if they both carry equal currents, i One Ampere is that current which, when flowing in each of two very long, straight, parallel wires, one meter apart, causes each wire to feel a force of attraction of 2x10 -7 Newtons per unit length. Coulomb is amount of charge which passes a surface in 1 sec if current through it is 1 A

Problem 4 Consider a very long (essentially infinite), tightly wound coil with n turns per unit length. This is called a solenoid. Assume that the lines of B are parallel to the axis of the solenoid and non-zero only inside the coil and very far away. Also assume that B is constant inside. Find B inside the solenoid if there is a current i flowing through it.

Problem 3 An infinitely long wire has 5 amps flowing in it. A rectangular loop of wire, oriented as shown in the plane of the paper, has 4 amps in it. What is the force exerted on the loop by the long wire?

Exercise 5

Consider the coaxial cable shown below. This represents an infinitely long cylindrical conductor carrying a current i spread uniformly over its cross section and a cylindrical conducting shell around it with a current i flowing in the opposite direction. The second i is uniformly spread over the cross section of the shell. Find magnetic field everywhere. c b ai

Induced EMF and Inductance 1830s Michael Faraday Joseph Henry

Faraday’s Law of Induction The induced EMF in a closed loop equals the negative of the time rate of change of magnetic flux through the loop

There can be EMF produced in a number of ways: A time varying magnetic field An area whose size is varying A time varying angle between and Any combination of the above

R From Faraday’s law: a time varying flux through a circuit will induce an EMF in the circuit. If the circuit consists only of a loop of wire with one resistor, with resistance R, a current Which way? Lenz’s Law: if a current is induced by some change, the direction of the current is such that it opposes the change.