Transcript small hans
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. B l i generator mv r qB The angular velocity v v qB r mv m qB Uniform magnetic field, vB Uniform B, v B 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 ( 1.60 1019 C, m 1.67 1027 kg) is placed in the uniform magnetic field directed along the x-axis with magnitude 0.500 T. Only the magnetic force acts on the proton. At t=0 the proton has velocity components vx 1.50 105 m / s, v y 0, vz 2.00 105 m / s. 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). B d S 0 Current carrying wires 1820 Hans Christian Oersted Hans Christian Ørsted Ampere’s Law B d r i 0 The field produced by an infinite wire 0 i B 2 a 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 dS i ( ds r ) dB 3 r i r 0 i (ds r ) dB 4 r3 0 is called permeability of free space 0 4 10 7 webers /( amp meter) 4 10 7 N /( amp) 2 (Also called Ampere’s principle) Problem 2 R 2R Field of a Current Carrying Loop r R x B 0i R2 2 (R2 x2 ) 3 (along x) 2 0 M x R B 2 x 3 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. m0 -7 N = 2× 10 2 2p A 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. b a c i 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 d B EMF dt d B E dr dt 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 B and dS Any combination of the above S S d B B dS B dS BdS cos 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 EMF i R 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. d B E dr dt