March 7 Physics 54 Lecture Professor Henry Greenside

Outline Chapter 28: Ampere’s law: deducing B for complicated currents that have a symmetry. Application of Ampere’s law to wires, solenoids and toroids. Reminder: we are skipping Sections 28-6, -7, -9, and -10. But please read Section 28-8 which has some useful and interesting qualitative information about solenoids.

Key Formulas from the Previous Lecture

PRS: Loop Pushed Toward Wire I I A closed rectangular loop of wire is pushed with constant speed toward a wire carrying a current I as shown. Then 1.Nothing happens to the loop. 2.A current flows clockwise in the loop. 3.A current flows counterclockwise in the loop. 4.I don’t know what to do here…

PRS: Current Loop Near Wire I1I1 A closed rectangular loop of wire carrying a clockwise current I 2 is near a vertical wire with current I 1. Then 1.The loop will move to the right. 2.The loop will move to the left. 3.The loop will move up. 4.The loop will move down. 5.The loop will rotate without its center moving. I2I2

At the Whiteboard 1.Detailed explanation of what happens when you push a loop toward a wire carrying a current: a current starts to flow in the loop and a net force appears that opposes the direction of the pushing. 2.A straight wire segment being pushed through a magnetic field acts like a battery with voltage difference vLB, where v is the speed of the wire, L is the length of the wire, and B is the magnitude of uniform magnetic field. 3.Discussion of Ampere’s law: the magnetic analog of Gauss’s law for deducing the magnetic field of complicated but symmetric currents. 4.Discussion of line integral on left side of Ampere’s law. 5.Applications of Ampere’s law to a single wire, to a hollow wire, and to a solenoid.

Clarification for Ampere’s Law: What is meant by “positive” and “negative” currents? Ampere’s law says that the line integral of the magnetic field along some imaginary (mathematical) closed loop is equal to some constant times the total current enclosed. But how do we choose a positive or negative sign for the currents that pass through the loop? The answer is to use the right hand rule on the loop: if in your mind you grasp the loop with your right hand so that your thumb points in the direction that your are tracing out the loop, then your fingers curl into the inside of the loop in the direction of positive currents. If you think about it, this makes sense since it gives the right answer when applied to a circular loop centered on a straight wire carrying a current I.

Solenoids: A Practical Way to Produce A Uniform Magnetic Field Will use Ampere’s law at whiteboard to derive this important relation. The quantity n is the ratio of the number of windings N per length L of the solenoid, I is the current that flows through each coil of the solenoid. Use the right hand rule for a single coil to deduce the direction of the magnetic field lines inside the solenoid.

Helmholtz Coils: A Quick and Easy Solenoid

PRS Question: Evaluation of a Line Integral A BC D x L E FG H

Practice With Ampere’s Law: B Field of A Current Sheet Hammer a wire into a thin planar sheet of large width L and thickness d, and pass a current I through the wire uniformly so that you have a constant “area current density” j=I/A=I/(Ld). 1.What direction is the magnetic field everywhere in space? 2.How strong is the magnetic field everywhere in space? Note: “Everywhere” means inside the sheet as well as outside the sheet