1. Wednesday(03/17): Intro. to Ch.30, Sources of the Magnetic Field and

the Biot-Savart Law. More Right-Hand-Rules.  HW: Read and Study pages

926-38, then solve prob. 1, 3, 5, 6, and 10 on pages 957-8.

2. Thursday(03/18): Ampere's Law and The Magnetic Field of Coils and

Solenoids. HW: Read and Study pages 938-47, then solve problems 13,

16, 21, and 26 on pages 958-60.

3. Friday(03/19): Lab on the Magnetic Field of a Coil. HW: Process Lab

Data and write Lab Report, due Tuesday.

4. Monday(03/22): Defining and Calculating Magnetic Flux. HW: Read

and Study pages 847-56, then solve problems  34, 37, 39, and 46 on

pages 961-2.

5. Tuesday(03/23): REVIEW for Chapter 30. HW: Complete All Review

Handouts.

6. Wednesday(03/24): TEST on Ch.30 - Sources of the Magnetic Field.

HAVE A SAFE AND RESTFUL SPRING BREAK. HW: Visit the web-site

for notes and plans for Ch.31 - Faraday's Law.

Website Notes for Ch.30: Sources of the Magnetic Field.

The objectives for Chapter 30 are: (1) DETERMINE both the magnitude and

the direction (using a right hand rule) of the magnetic field produced by a

point charge moving at constant velocity, (2) UNDERSTAND and KNOW HOW

TO USE the Biot-Savart law to determine the magnetic field of a current

element, (3) UNDERSTAND and USE CORRECTLY Ampere's law and symmetry

considerations for magnetic field calculations from a given current distribution,

(4) DETERMINE the force (both magnitude and direction) between long parallel

conductors, and (5) DERIVE, using Ampere's law, both the magnitude and

direction (another right hand rule) of the magnetic fields due to (a) a long,

straight, current-carrying conductor; (b) idealized solenoid-like devices; and

(c) long cylindrical conductors with uniform current distributions.

Points to Remember:

1. Magnetic fields arise from charges, similarly to electric fields, but are

different in that the charges must be moving.

2. A long straight wire carrying a current is the simplest example of a moving

charge that generates a magnetic field. We mentioned that the direction of the

force a charge felt when moving through a magnetic field depended on the

right-hand rule.

3. The direction of the magnetic field due to moving charges will also depend

on the right hand rule. For the case of a long straight wire carrying a current, I,

the magnetic field lines wrap around the wire. By pointing one's right thumb

along the direction of the current, the direction of the magnetic field can by

found by curving one's fingers around the wire.

4. The strength of the magnetic field depends on the current I in the wire and

r, the distance from the wire. The equation is B=μ_oI∕(2πr) , with the constant

μ_o, "mu naught", given as 4π x 10^-7 Tm/A .

5. The constant is the permeability of free space. The reason it does not appear

as an arbitrary number is that the units of charge and current (coulombs and

amps) were chosen to give a simple form for this constant.

6. If one remembers the case of the electric field of a uniformly charged wire,

it also fell off as 1/r. There is no real analogy to Coulomb's law for magnetism,

as the magnetic field of a point charge is complicated since it can't be standing

still to generate a magnetic field.

7. Ampere's law is the magnetic equivalent of Gauss's law. It is different in that

it refers to a closed loop and the surface enclosed by it (rather than a closed

surface and the volume enclosed by it, as is the case with Gauss's Law).

Consider a closed loop, not necessarily a circle, which is broken into small

elements of length dl , with a magnetic field dB at each element.

8. The sum over elements of the component of the magnetic field along the

direction of the element, times the element length, is proportional to the

current I that passes through the loop.

9. This is Ampere's law. "The line integral ∫B·dl=μ_oI ." For the case of a wire,

the loop can be a circle drawn around the wire, and since the field is always

tangent to the circle, cos(θ) = 1. The circumference of the circle of radius r is

2πr, therefore Ampere's law becomes: B·2πr = μ_oI .

10. Ampere's law will also allow us to calculate the magnetic field for a

solenoid. A solenoid is a coil of wire designed to create a strong magnetic field

inside the coil. By wrapping the same wire many times around a cylinder, the

magnetic field due to the wires can become quite strong.

11. Consider a solenoid of N turns, or loops. Intuitively we know that more

loops will bring about a stronger magnetic field. Ampere's law can be applied

to find the magnetic field inside a long solenoid as a function of the number of

turns per unit length, n = N/L , and the current, I. The equation we get is given by,

B=μ_o(N/L)I .

12. Only the upper portion of the path contributed to the sum because the

magnetic field is zero outside, and because the vertical paths are perpendicular

to the magnetic field.

13. This is the result we have been after. The magnetic field inside a solenoid is

proportional to both the applied current and the number of turns per unit length.

There was no dependence on the diameter of the solenoid or even on the fact

that the wires were wrapped around a cylinder and not a rectangular shape. Most

importantly, the result did not depend on the precise placement of the path inside

the solenoid, indicating that the magnetic field is constant inside the solenoid.

14. A current-carrying wire acts as a source of magnetic field. A second wire will

feel a force from the magnetic field of the first one. The force on wire a due to

wire b will always be equal and opposite to the force on wire b due to wire a.

Using the right-hand rule one can show the following: (a) Parallel wires with

current flowing in the same direction, attract each other. (b) Parallel wires with

current flowing in the opposite direction, repel each other.

15. Biot-Savart Law: The magnetic equivalent of Coulomb's law is the Biot-Savart

law for the magnetic field produced by a short segment of wire, dl, carrying current

I: dB = (μ_oI/4π) dl sin(φ)/r² , where the direction of dl is in the direction of the

current and where the vector , r , points from the short segment of current to the

observation point where we are to compute the magnetic field. Since current must

flow in a circuit, integration is always required to find the total magnetic field at

any point. The constant , μ_o ,is chosen so that when the current is in amps and

the distances are in meters, the magnetic field is correctly given in units of Tesla.

16. A quick comparison of this value with the Biot-Savart law probably makes you

wonder what role 4π is supposed to play here. It plays the same role it did in

Coulomb's law: it was required in Coulomb's law so that Gauss's law wouldn't

have a 4π , and it is required in the Biot-Savart law so that Ampere's law won't

have one either.

17. There are two simple cases where the magnetic field integrations are easy to

carry out, and fortunately they are in geometries that are of practical use. We

use the formula for the magnetic field of an infinitely long wire whenever we want

to estimate the field near a segment of wire, and we use the formula for the

magnetic field at the center of a circular loop of wire whenever we want to

estimate the magnetic field near the center of any loop of wire.

18. Infinitely Long Wire: The magnetic field at a point a distance r from an

infinitely long wire carrying current I has magnitude B = μ_oI/(2πr) , and its

direction is given by a right-hand rule: point the thumb of your right hand in the

direction of the current, and your fingers indicate the direction of the circular

magnetic field lines around the wire.

19. Circular Loop: The magnetic field at the center of a circular loop of current-

carrying wire of radius R has magnitude B = μ_oI/(2R) , and its direction is given

by another right-hand rule: curl the fingers of your right hand in the direction of

the current flow, and your thumb points in the direction of the magnetic field

inside the loop.

20. Long Thick Wire: Imagine a very long wire of radius a carrying current I

distributed symmetrically so that the current density, J, is only a function of

distance r from the center of the wire. Ampere's law can be used to find the

magnetic field at any radius r. Outside the wire, where r >= a, we have,

B = μ_oI/(2πr) , just as if all the current were concentrated at the center of the

wire. Inside the wire, where r < a, we have B = μ_oI(r)/(2πr) , where I(r) is the

current flowing through the disk of radius r inside the wire; the current outside

this disk contributes nothing to the magnetic field at r. Note that this is

analogous to the result for symmetric electric fields, discussed in Chapter 24.

21. Long Solenoid: Imagine a long solenoid of length L with N turns of wire

wrapped evenly along its length. Ampere's law can be used to show that the

magnetic field inside the solenoid is uniform throughout the volume of the

solenoid (except near the ends where the magnetic field becomes weak) and

is given by B = μ_o(N/L)I = μ_onI , where n = N/L.

22. Toroid: Imagine a toroid consisting of N evenly spaced turns of wire

carrying current I. (Imagine winding wire onto a bagel, with the wire coming

up through the hole, out around the outside, then up through the hole again,

etc..) Ampere's law can be used to show that the magnetic field within the

volume enclosed by the toroid is given by B = μ_oNI/(2πR) .

And still, we need these steps to solve any problem in Physics:

(i) read the problem and identify the given variables

(ii) determine what you are asked to solve for

(iii) find the correct equation to use

(iv) use Algebra, Trigonometry, and/or Calculus to isolate the unknown

(v) substitute-in the given information and simplify.

Click Here To Check your Homework Solutions.

12.6 μT	11.3 GVm/s, .100 A	56 μT
9.27x10-24 Am2, down	-BπR2cosθ, BπR2cosθ	12.5 T
4.5μ0 I / πL, stronger	28.3 μT, into	24.7 μT, into
μ0 I / 2πx, out of	20μT toward bottom of page	3.60 T, 1.94 T
(1/4+1/π)μ0 I / 2r, into	μ0 I / 4πx, into	(1+1/π)μ0 I / 2R, into
10μT out of, 80μT toward, 16μT into, 80μT toward

THE AP PHYSICS C ARCHIVES Ch.1: Physics Intro. Ch.2: Linear Motion. Ch.3: Vectors. Ch.4: 2-Dim Motion. Ch.5&6: Newton's Laws. Ch.7&8: Work&Energy. Ch.9: Momentum. Ch.10&11: Rotary Motion. Ch.12: Elasticity. Ch.13: Gravitation. Ch.15: SHM. Mechanics Review. Ch.23: Electric Fields. Ch.24: Gauss's Law. Ch.25: Electric Potential. Ch.26: Capacitance. Ch.27: Current/Resistance. Ch.28: DC Circuits. Ch.29: Magnetic Effects.