For Physics 1320, April 17, 2021

Preliminary Comments on Michael Faraday:

From: http://bookbinding.com/short-essays-on-bookbinding/alias.html

MR. FARRADAY, now aged forty-two, at the head of one of the noblest of the sciences, honoured as the compeer of Cuvier, Laplace, and Buckland, was the son of a poor blacksmith, and was apprenticed at nine years of age, to a bookbinder in Blandford street, and earned his bread by that calling until he was twenty-two! Mr. McGrath, now secretary to the Athenceum, happening five-and-twenty years ago, to enter the shop of Ribeau, observed one of the men zealously studying the book he ought to have been binding. He approached - it was a volume of an Encyclopedia, open at "Electricity." He entered into conversation with the journeyman, and was astonished to find him a self-taught chemist of no slender dimensions. He presented him with a set of tickets for Davy's Lectures at the Royal Institution ; and daily, thereafter, might he be seen, pen in hand, just over the clock, opposite the chair. At last the course terminated; but Farraday's spirit had received a new impulse, which nothing but dire necessity could have restrained; and from that he was saved by the promptitude with which, on his forwarding a modest outline of his history, with the notes of these lectures, to Davy, that great and good man rushed to the rescue of kindred genius. Sir Humphrey immediately appointed him an assistant in the laboratory; and after two or three years had passed, he

found Farraday qualified to act as his secretary. The steps of his subsequent progress are well known; he travelled over the continent with Sir Humphrey and Lady Davy, and he is now what Davy was when he first saw Davy.

Editor's note, from wikipedia:

The young Michael Faraday, one of four children, having only the most basic of school educations, had to largely educate himself. At fourteen he became apprenticed to a local bookbinder and bookseller George Riebau and, during his seven-year apprenticeship, he read many books, including Isaac Watts's The Improvement of the Mind, and he enthusiastically implemented the principles and suggestions that it contained. He developed an interest in science, especially in electricity. In particular, he was inspired by the book Conversations in Chemistry by Jane Marcet.

Preliminary Comments on Joseph Henry by John Rigden of the American Physical Society

http://www.aps.org/programs/outreach/history/historicsites/henry.cfm

Joseph Henry

The Albany Academy, Albany, NY On April 27, 2007, John Rigden, Chair of the APS Historic Sites Committee, presented the Albany Academy in Albany, NY with a plaque to honor physicist Joseph Henry for his pioneering work performed there on electromagnetism, in particular the discovery of self-inductance. Henry was acknowledged as the inventor of the electric motor, the father of daily weather forecasts, and the preserver of the Smithsonian. He was so revered that the government closed for his funeral on May 16, 1878, a funeral attended by the President, Vice-President, the Cabinet, the members of the Supreme Court, Congress, and the senior officers of the Army and the Navy. During his time, Henry was thought of as the successor to Benjamin Franklin in his experiments on electromagnetism. His scientific career began when he picked up and read the book Popular Lectures on Experimental Philosophy, Astronomy, and Chemistry when he was sixteen. Fascinated by this glimpse of science, he resolved to learn more.

Joseph Henry

Most of Henry’s research and contributions to science occurred at Albany Academy, where he enrolled in 1819. In his day it offered him the equivalent of a college education though we would see it today as closer to secondary school. Henry went beyond the coursework, avidly reading books in every area of science and many other fields. He supported himself by working as a schoolmaster then tutor, road surveyor to finally Professor of Mathematics and Natural Philosophy at the Albany Academy. At the Albany Academy, his research centered on experiments with magnetism. He was the first to coil insulated wire tightly around an iron core in order to make an extremely powerful electromagnet. Using this technique, he built the most powerful electromagnet at the time for Yale. He also showed that, when making an electromagnet using just two electrodes attached to a battery, it is best to wind several coils of wire in parallel, but when using a set-up with multiple batteries, there should be only one single long coil. The latter made the telegraph feasible. He continued to improve on devices with his research and, in 1831, created one of the first machines to use electromagnetism for motion. This was the earliest ancestor of modern DC motor. It did not use rotating motion for power, but was merely an electromagnet perched on a pole, rocking back and forth. In the relatively isolated environment of the Albany Academy, Henry discovered the fundamental property of self-inductance. However, the British scientist Michael Faraday discovered it as well at about the same time, and, being first to publish his results, became the officially recognized discoverer of the phenomenon. Henry left the Albany Academy for Princeton in 1832, and then became the Secretary of the brand new Smithsonian Institution in 1846. His job was to give scientific advice to the government and to make sure the Institution remained a research center as well as a museum. In the spring of 1863 Henry became one of the founding members of the National Academy of Science, and served as Academy president beginning in 1867. He served as both the National Academy of Science president and secretary of the Smithsonian Institution until his death in 1878. In 1893 his name was given to the standard electrical unit of inductive resistance, the henry.

Faraday’s Law of Induction 1831 (Henry 1832)

Faraday’s experimental discovery applies to closed loops of wire. When you change the magnetic flux through the loop of wire, you cause a current to flow in the wire. The magnetic flux is just like the electric flux we are familiar with, except that you use B instead of E.

You have to integrate B over the surface area bounded by the loop of wire. The dot product is between

the field B and the area A, and the vector over the A

is the usual thing – we assign to the area a vector that points normal to the surface of the loop.

If the magnetic field is uniform in space, then you don’t have to do an integral, and you can write this as the dot product between B and A:

Experiments showed that the product of the resistance in the wire loop and the current is equal to the time rate of change of the flux.

The faster you change the flux, the greater the current. Of course, we know Ohm’s law, and we know that charges at rest only move in a loop of wire if you have a nonvanishing electric field, so using the fact that

we can rewrite Faraday’s law in the form,

But wait. Why are we using the ancient “EMF” (electromotive force) notation instead of just using voltage in this expression? Also, why have we tossed in that negative sign? We are using EMF to express the fact that the electric field produced in the loop is unlike anything we have dealt with before. If you take the line integral of the electric field around the closed loop of wire,

you don’t get zero, as you would for example if there were a battery hooked up in series in the closed loop. Indeed, the electric field produced in the wire is a nonconservative field. We have not encountered a nonconservative field before this. In fact, we had thought that such a thing doesn’t exist in nature. Energy-wise it makes no sense. If a charge travels around the wire in the direction of the field and returns to its starting point, its energy decreases. If it travels around the loop a second time in the same direction, its energy decreases even more. If it keeps going around the loop in the same direction, its potential energy continues to decrease. There is apparently no end to this energetic “death spiral”.

Isn’t this a violation of Kirchoff’s voltage law, the law saying that the sum of the voltages around a closed loop has to be zero? Yes, it is.

Why is there a minus sign? There is real meaning to this minus sign, but to use this properly you have to be exacting in how you define the EMF, to make it compatible with the way you define the area vector pointing out of your wire loop. It is true that you have the freedom to choose the direction in which you take the integral of E dot d ,

in the wire loop. I show two directions for defining in the figure below, clockwise and ccwise.

Using the right hand screw rule, when the direction of integration is taken to be clockwise, the area vector points into the page; when the direction of integration is counterclockwise, the area vector points out of the page. There is no right or wrong to this – you can choose either direction to define EMF, but keep in mind that when you make your choice, you are also choosing a direction for the area vector. If the electric field happens to be encircling the wire in a clockwise direction, then the EMF you calculate

when you travel clockwise around the loop will be positive since E d is positive when the two

vectors E

and d are in the same direction. For the same case, the EMF you calculate when you go

counterclockwise around the loop will be negative because the two vectors E

and d are in opposing

directions. The direction of the area vector determines the sign of the dot product between B and A. If B

is pointing into the page, then when A is also pointing into the page, B A

will be positive, and when A

is pointing out of the page, B A

will be negative. If B is pointing out the page, then when A is pointing

into the page, B A

will be negative, and when A is pointing out of the page , B A

will be positive. All of these directions and their associated signs are linked together by the expression in the box above, and there is no ambiguity.

So look again at the expression that we refer to as Faraday’s law, and think about the fact that the negative sign is important – realize that hidden within the symbols on the left, and the right, are particular directions of integration, directions that are linked to one another by the right hand rule.

To see how this works, let us consider an example. Let us suppose the magnetic field B is directed into the plane of the paper, as shown by the red arrow in the figure. Suppose also that the magnitude of B is increasing in time. What is the direction of the induced electric field?

(i) Consider the left hand drawing first. The vectors B and A are parallel, so the flux is

positive, and if B is increasing then the flux is increasing. This means that d

dt is

positive, and since d

dt

E - , it follows that E is negative. But if E is negative, then

it means that the induced nonconservative electric field E

points around the loop in the

counterclockwise direction because the dot product with d has to be negative if the

EMF is negative.

(ii) Consider the right hand drawing. Here the vectors B and A are antiparallel, so the flux is

negative. If B is increasing, the flux is becoming more negative. This means that d

dt is

negative, and since d

dt

E - , it follows that E is positive. But if E is positive, then

it means that the induced nonconservative electric field E

points around the loop in the

counterclockwise direction because we need its dot product with d to be positive in

order to get a positive EMF.

In summary, in both cases (i) and (ii), we conclude that the nonconservative electric field is induced in the wire loop in the counterclockwise direction.

Lenz’s Law 1834

In 1834 Heinrich Lenz came up with a much easier way to keep track of the direction of the induced current in Faraday’s law. The so-called “Lenz’s Law” is stated as follows: “The current will always be induced in such a manner as to oppose the change in flux.”

Let us consider the same example of the B increasing through the loop, as depicted above. Because the flux is increasing, we know there will be an induced current. According to Lenz’s law, the current will be induced to “oppose the change”. What is the meaning of “oppose”? There are several ways to think of this, but one way is to realize that when the current is induced in the wire loop, the wire loop becomes a small electromagnet, with a North pole on one side and a South pole on the other. In the drawing, we see that the external field has its North pole coming into the loop. Since the field

strength is increasing, the externally-imposed North pole is increasing. The loop will “do its best” to negate this – by sticking it’s North pole in the opposite direction.

Suppose we have the same situation, with the external field in the same direction, but the external field is decreasing. Now the current in the loop will be induced so as to sustain the diminishing field. This means the current will go in the other direction, so that the North pole of the induced current loop aligns itself with the North pole of the external field, as though in a feeble attempt to keep the field from diminishing.

Numerical Example:

Consider the figure above. Suppose that the external field increases at a constant rate of 1 Tesla per second. Suppose that the area of the loop is 2 square meters. Suppose that the wire has a resistance of 5 Ohms. What will be the magnitude and direction of the current produced in the loop?

(a) First, as discussed above, the current will go in the counterclockwise direction, so that the magnetic field that arises in the loop is directed opposite the external magnetic field, as if it is attempting to prevent the increase.

(b) The magnitude of the EMF is given by the time rate of change of the flux. Since the B is increasing linearly in time, we can write

Second Example: An AC Generator

Suppose that we have a square loop of wire with 1000 turns and area of 1 square meter that rotates about an axis through its center in a uniform magnetic field of 1 Tesla at a rate of 60 revolutions per second (60 Hz). What is the EMF produced in the loop as a function of time?

In this example we haven’t paid a lot of attention to the direction of the induced current (i.e. the direction of the EMF) because it alternates between positive and negative. It is interesting to think about how the loop has to be oriented when we reach the points where the EMF is zero, or where the EMF is maximum. There must be particular orientations where, instantaneously, the flux isn’t changing at all.

If the generator is connected to a resistor (i.e. if we want to “light a city”), we should imagine putting this voltage across a resistor.

As far as lighting is concerned, we are interested in the energy used over the long haul, so we care about the time-average of the power.

We have encountered averages of this sort before. The average of the square of a quantity is called the “mean-square”. The square root of the average of the square of a quantity is called the RMS, “root mean-square”. For sinusoidal AC power generation, you can see that the time average power is given by

where the RMS voltage is given by

In the US, the RMS voltage in a home is about 120 Volts, and this implies that the maximum is about