Feynman Lectures Simplified 2B: Magnetism & Electrodynamics

By Robert Piccioni

Feynman Simplified: 2B covers one quarter of Volume 2 of The Feynman Lectures on Physics. The topics we explore include:

• Magnetostatics
• Dynamic Electric & Magnetic Fields
• Filters & Transmission Lines
• Electromagnetic Waves in Vacuum
• Electrical Circuits & Components
• Circuit & Cavity Resonances

And if you are looking for information about a specific topic, peruse our free downloadable index to the entire Feynman Simplified series found on my website "Guide to the Cosmos . com"

• Language: English
• Publisher: Robert Piccioni
• Release date: Apr 28, 2017
• ISBN: 9781370073375

Robert Piccioni

Dr Robert Piccioni is a physicist, public speaker, educator and expert on cosmology and Einstein's theories. His "Everyone's Guide Series" e-books makes the frontiers of science accessible to all. With short books focused on specific topics, readers can easily mix and match, satisfying their individual interests. Each self-contained book tells its own story. The Series may be read in any order or combination. Robert has a B.S. in Physics from Caltech, a Ph.D. in High Energy Physics from Stanford University, was a faculty member at Harvard University and did research at the Stanford Linear Accelerator in Palo Alto, Calif. He has studied with and done research with numerous Nobel Laureates. At Caltech, one of his professors was Richard Feynman, one of the most famous physicists of the 20th century, and a good family friend. Dr. Piccioni has introduced cutting-edge science to numerous non-scientific audiences, including school children and civic groups. He was guest lecturer on a National Geographic/Lindblad cruise, and has given invited talks at Harvard, Caltech, UCLA, and Stanford University.

Book preview

Feynman Simplified

2B: Magnetism

& Electrodynamics

Everyone’s Guide

to the

Feynman Lectures on Physics

by

Robert L. Piccioni, Ph.D.

Second Edition

Copyright © 2016

by

Robert L. Piccioni

Published by

Real Science Publishing

3949 Freshwind Circle

Westlake Village, CA 91361, USA

Edited by Joan Piccioni

V1610111

All rights reserved, including the right of

reproduction in whole or in part, in any form.

Visit our web site

www.guidetothecosmos.com

Everyone’s Guide to the

Feynman Lectures on Physics

Feynman Simplified gives mere mortals access to the fabled Feynman Lectures on Physics.

This Book

Feynman Simplified: 2B covers one quarter of Volume 2 of The Feynman Lectures on Physics. The topics we explore include:

Magnetostatics

Dynamic Electric & Magnetic Fields

Filters & Transmission Lines

Electromagnetic Waves in Vacuum

Electrical Circuits & Components

Circuit & Cavity Resonances

To learn more about the Feynman Simplified series, to receive updates, and send us your comments, click here. 

To further Simplify your adventure, learn about my Math for Physicists that explains the math to master Feynman physics.

Looking for information about a specific topic? Peruse our free downloadable index to the entire Feynman Simplified series.

If you enjoy this book, please do me the great favor of rating it on Amazon.

Table of Contents

Chapter 14: Magnetostatics

Chapter 15: The Vector Potential

Chapter 16: Electromagnetic Statics Wrap-Up

Chapter 17: Induced Currents

Chapter 18: Exploring Induction

Chapter 19: Maxwell’s Equations

Chapter 20: Wave Solutions in Vacuum

Chapter 21: Solutions with Charges

Chapter 22: Electrical Circuits

Chapter 23: Cavity Resonators

Chapter 24: Review: Feynman Simplified 2B

Chapter 14

 Magnetostatics

In Feynman Simplified 2A, Section §4.2, we learned that in static conditions, when charges and currents do not change over time, Maxwell’s four equations reduce to two de-coupled pairs of equations:

Electrostatics:

Ď • E = ρ / ε0

Ď × E = 0

Magnetostatics:

Ď × B = j / ε0

Ď • B = 0

In the static case, we can treat electricity and magnetism as if they were unrelated phenomena. They become interconnected only when charges accelerate.

The preceding nine chapters explored electrostatics. We now begin the study of magnetostatics, the physics of static magnetic fields.

§14.1 Magnetism

Magnetic forces are more complex than electric forces, as demonstrated by the Lorentz force that combines both effects.

F = q (E + v × B)

The electric component of force depends only on the charge q and the electric field E. The magnetic component depends on charge q, magnetic field B, and also on the charge’s velocity v, both on its magnitude and its orientation.

The magnetic component of force F is always perpendicular to both v and B. Its polarity is defined by the cross product and the right hand rule. Please ensure you master this sign convention; it would be a shame to solve a complex problem and get the wrong answer because you used the wrong fingers. Different people get the right answer in different ways; this image shows how I do it. 

With my index finger pointing along the first vector (v in this case), and my middle finger pointing along the second vector (B), my the thumb points along the cross product v×B. For q>0, that is the direction of F. Sometimes, you may need to twist your wrist awkwardly.

Figure 14-1 shows vectors v and B within the same plane (shown in gray), with θ being the angle between them.

Figure 14-1 Magnetic Force: F = qv×B

Force F is perpendicular to both vectors and has magnitude:

F = q v B sinθ

Magnetic fields can be measured in many equivalent units. For θ=90º, v=1m/s, q=1 coulomb, and F=1 newton, the corresponding value of B is:

1 newton-second / coulomb-meter

1 volt-second / meter²

1 tesla = 1 weber / meter² = 1 kg / amp sec²

1 tesla = 10,000 gauss

§14.2 Conservation of

Current & Charge

Electric current is electric charge in motion. A single charge q with velocity v produces a current J given by J=qv. Like fields, currents from multiple charges add vectorially according to the principle of linear superposition.

In the same way that we dealt with heat flow in Feynman Simplified 2A, Section §2.3, Feynman defines current density as a vector field j(r). At each point r, j points in the direction of current flow, and the magnitude of j equals the amount of charge passing parallel to j per unit area per unit time. The amount of charge ΔQ passing in time Δt through a small surface of area S, whose unit normal is n, equals:

ΔQ / Δt through S = j • n S

This equation is illustrated in Figure 14-2. 

Figure 14-2 Current Flow Through S

Figure 14-2 shows a 2-D cross section of a physical situation that is three-dimensional. Here, current j is vertical and we represent surface S with a bold diagonal line whose unit normal n is at angle θ to j.

If a collection of electric charges has charge density ρ(r) per unit volume, and average velocity v, each charge moves an average distance vΔt in time Δt. In Figure 14-2, all charges within the gray parallelepiped are below S by a distance of no more than vΔt; hence all those charges move up through S during time Δt. The amount of charge ΔQ contained within the gray parallelepiped is its volume V multiplied by ρ. This is:

ΔQ through S = (V) ρ

ΔQ through S = (v Δt S cosθ) ρ

ΔQ / Δt through S = v • n S ρ

ΔQ / Δt through S = j • n S

with j = v ρ

If the collection of charges consists of N identical particles per unit volume, each with charge q, the last equation becomes:

j = v N q

The total electric current J, the total charge per unit time, flowing through any surface S equals the current density integrated over S:

J = ∫S j • n dS

Physicists generally use i or I for current, but that is harder to read in an eBook than j or J (is l capital i or lower case L or the number 1?). Also i may be confused with √–1. For clarity, I use j and J for electrical current.

With Gauss’ law, we can replace the above surface integral with an integral over the volume V contained within a closed surface S:

J = ∫V Ď • j dV

J is then the current flowing out of V. A fundamental principle of physics is that net electric charge is locally conserved.

When an electron and an antielectron (a positron) annihilate, both particles disappear, their mass energy converts into gamma rays, and one negative and one positive charge vanish. An electron / positron pair can also be created, for example in neutral pion decay, with the creation of one negative and one positive charge. Neither reaction changes net charge, the sum of all positive charges reduced by the sum of all negative charges. Physicists have never observed a reaction that fails to conserve net charge.

Local conservation is a stronger requirement than global conservation. If charge were conserved only globally, an electron could disappear in London provided that a new electron appeared in Auckland. Local conservation states that the net charge within any volume V changes only when a current crosses its boundary. If the current carries charge ΔQ into V, the net charge in V increases by ΔQ, which can be positive or negative.

It is the principle of local charge conservation that leads us to focus on how electric currents carry charge from place to place. With local charge conservation we are assured that:

dQ/dt within V = ∫V dρ/dt dV = – ∫V Ď • j dV

(Recall that Q within V = ∫V ρ dV). Since the above equation must be true for every volume V, this proves the differential equation:

dρ/dt = – Ď • j

§14.3 Magnetic Force

on Currents

In Feynman Simplified 2A, Section §1.7, we said magnetic fields exert forces on wires carrying currents. We can now derive an equation for that force. The Lorentz force law provides the force on an individual charge q moving with velocity v in magnetic field B:

F = q v × B

Let N be the number of charges per unit volume, each with charge q moving with velocity v. In field B, let’s calculate the force G on a short piece of cylindrical wire of length ΔL and cross sectional area S. The force F on each individual charge q is perpendicular to both B and v, with v parallel to the wire’s axis, the direction of current flow, as shown in Figure 14-3.

Figure 14-3 Magnetic Force F on Charge q

By linear superposition, the total force per unit volume equals NF, (the number of charges per unit volume) multiplied by (the force per charge). The total force G is then NF multiplied by SΔL, the volume of the piece of wire. This means:

G = N (q v × B) S ΔL

Recall that j = v N q, and that the total current J equals the integral of j across the surface through which current passes. For a wire with uniform current density: J=jS. Combining all this yields:

G = j × B S ΔL

G / ΔL = J × B

The force on the wire per unit length is G/ΔL that equals J×B.

As Feynman stresses in V2p13-3, the force on a wire is proportional to the total current and not on q, the charge of individual particles. We get the same force for positive charges moving right as for negative charges moving left. This is fortunate, since scientists originally incorrectly guessed the carrier polarity.

§14.4 Magnetic Fields From

Moving Charges

Also in Feynman Simplified 2A, Section §1.7, we said current-carrying wires exert forces on magnets. Let’s see how that happens.

Newton’s third law of motion, action-begets-reaction, suggests that if a magnet exerts a force on a current-carrying wire, there should be a reaction force exerted on the magnet, the source of the original force. Indeed, from our experience with magnets, we know that they exert non-contact forces upon one another through their magnetic fields.

For constant fields and currents, the equations of magnetostatics are:

Ď • B = 0

c² Ď × B = j / ε0

Feynman pauses here to note that statics — the condition of constant fields and currents — is an idealization, similar to friction-less mechanical motion. We make such approximations to facilitate learning physical principles by minimizing distracting complications. If you are going to design a rocket to go to Mars, you should make more realistic assumptions. If you want to learn physics in order to qualify for that job, read on.

Feynman notes that since Ď•Ď×ξ=0 for any vector field ξ, the last equation is valid only if Ď•j=0. From our earlier equation for the local conservation of charge we have dρ/dt=–Ď•j. Thus the equations of magnetostatics require dρ/dt=0. A changing charge density would generate changing electric fields, contrary to our static assumption of constant fields. All this means our assumptions are consistent, even if perhaps highly idealized.

A final comment in this regard: Ď•j=0 means that currents must flow in closed loops. In statics, currents cannot start here and end there; they must flow through complete circuits that might include batteries, inductive coils, and similar elements, but they cannot include capacitors. Constant currents change a capacitor’s charge and therefore its electric field.

We will get to electromagnetic dynamics in later chapters. Here we examine the physics of constant fields and currents.

The first equation of magnetostatics, Ď•B=0, says there are no magnetic monopoles: no magnetic equivalent of a single electric charge. We said earlier that a magnet’s B field lines start at its north pole and circle around to its south pole. But actually, the field lines continue within the magnet forming a closed loop. Magnetic field lines never start at point X or end at point Y. Feynman says that in some complex situations, the B field lines are not simple loops, but that they nonetheless never stop or start at a point. Everything in this paragraph is true for all of electromagnetism, static or dynamic.

The second equation of magnetostatics, Ď×B=j/c²ε0, enables calculating the B field produced by a current j. Using Stokes’ theorem we can relate the circulation of B to the flux of its curl:

∫S (Ď × B) • n dS = ∫Γ B • ds

Here, S is a surface enclosed by the curve Γ, n is the unit normal to S at each point, and ds is an infinitesimal tangent vector to Γ at each point. This is illustrated in Figure 14-4. 

Figure 14-4 Loop Γ & Surface S 

Substituting j/c²ε0 for the curl of B yields:

∫S (j / c²ε0) • n dS = ∫Γ B • ds

Thus (the tangential component of B integrated around curve Γ) equals (the total current flux through surface S) divided by (c²ε0). The equation, known as Ampere’s law, is:

∫Γ B • ds = current J through S / (c² ε0)

Comparing the equations of electrostatics with those of magnetostatics, we see a reversal of the roles of curl and divergence. For electrostatics, the curl of E is always zero, while the divergence of E is proportional to charge density. For magnetostatics, the divergence of B is always zero, while the curl of B is proportional to current density.

§14.5 Fields From Wires & Solenoids

Our first example of employing Ampere’s law is finding the magnetic field B around an infinitely long, cylindrical wire carrying current J.

We define a cylindrical coordinate system with the z-axis along the length of the wire, r being the radial distance from the z-axis, and β being the azimuthal angle around the wire, as shown in Figure 14-5.

Figure 14-5 Magnetic Field Near Wire

The dashed circle labeled Γ has radius r, is centered on the wire, and lies in a plane perpendicular to the wire.

In cylindrical coordinates, we can express the most general form of the B field as:

B = (R, ß, Z)

In general, B‘s components R, ß, and Z may be functions of all three coordinates r, β, and z. But due to the wire’s symmetry, none can be a function of β or z. We can see this by imagining rotating the wire around its axis, or moving the wire along its length. Neither action makes any physical difference due the wire’s infinite length and rotational symmetry. Hence no real physical entity, including the components of the magnetic field, can change as a function of β or z. Each component of B must be a function of r only.

In V2p13-5, Feynman says:

We will assume something which may not be at all evident, but which is nevertheless true: that the field lines of B go around the wire in closed circles.

We prove this in Section §15.3. This means B always points in an azimuthal direction, so R=Z=0 and:

B = (0, ß(r), 0)

Let’s calculate ß(r), the tangential component of B, on the circle Γ of radius r. Every point along Γ has the same r and therefore the same ß(r). By Ampere’s law:

∫Γ B • ds = J / c²ε0

2π r ß(r) = J / c²ε0

ß(r) = J / 2πε0 c² r

The tangential component ß is orthogonal to both J and the radial coordinate r. We can express this equation in vector form as:

B = J × r / 2πε0 c² r²

You might check that this cross product employs the right hand rule.

Since current-carrying wires produce magnetic fields, and since magnetic fields exert forces on one another (as we know from playing with magnets), it follows that current-carrying wires exert forces on one another. In Feynman Simplified 2A, Section §1.7, we said that parallel wires are pushed together if their currents are in the same direction and pushed apart if their currents are in opposite directions. This is illustrated in Figure 14-6.

Figure 14-6 Forces on Conductors

Here, current flows toward you in the central three wires, and flows away from you in the outer two wires. This is indicated using a standard convention: the black dots represent the tips of arrows pointing toward you, while the crosses represent the feathers of arrows pointing away from you.

Figure 14-6 shows two magnetic field lines (labeled B) produced by the central wire, and the four forces (labeled F) that these fields exert on the other four wires. As the figure demonstrates, the central wire attracts wires with parallel currents and repels wires with antiparallel currents. You might exercise