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(III) Two positive charges $+ Q$ are affixed rigidly to the $x$ axis, one at $x = + d$ and the other at $x = - d .$ A third charge $+ q$ of mass $m ,$ which is constrained to move only along the $x$ axis, is displaced from the origin by a small distance $s \ll d$ and then released from rest. $( a )$ Show that (to a good approximation) $+ q$ will execute simple harmonic motion and determine an expression for its oscillation period $T .$ (b) If these three charges are each singly ionized sodium atoms $( q = Q = + e )$ at the equilibrium spacing $d = 3 \times 10 ^ { - 10 } \mathrm { m }$ typical of the atomic spacing in a solid, find $T$ in picoseconds.

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A. $2 \pi \sqrt{\frac{m \pi \varepsilon_{0} d^{3}}{Q q}}$B. $0.2 \mathrm{ps}$

08:24

Shital Rijal

Physics 102 Electricity and Magnetism

Chapter 21

Electric Charge and Electric Field

Gauss's Law

Ngh?A N.

September 29, 2020

Q*q= - number ?

University of Michigan - Ann Arbor

Simon Fraser University

University of Sheffield

University of Winnipeg

Lectures

11:53

In physics, a charge is a physical property of matter that causes it to experience a force when placed in an electromagnetic field. Electric charge is a characteristic property of many subatomic particles, which determines their electromagnetic interactions. Electrically charged matter is influenced by, and produces, electromagnetic fields. The interaction between a moving charge and the electromagnetic field is the source of the electromagnetic force, which is one of the four fundamental forces.

18:38

In physics, electric flux is a measure of the quantity of electric charge passing through a surface. It is used in the study of electromagnetic radiation. The SI unit of electric flux is the weber (symbol: Wb). The electric flux through a surface is calculated by dividing the electric charge passing through the surface by the area of the surface, and multiplying by the permittivity of free space (the permittivity of vacuum is used in the case of a vacuum). The electric flux through a closed surface is zero, by Gauss's law.

0:00

(III) Two positive charges…

05:47

Two positive point charges…

26:10

1. Two pointcharges Q1…

06:54

Two particles $A$ and $B$,…

03:09

1wo equal negative charges…

12:36

A particle of mass $m$ and…

14:21

Two identical particles, e…

09:13

Four identical point charg…

here, we're going to examine the behavior of a small charge caught in between two equal charges but constrained to move along the line joining them. Um and we'll see what happens when that small charge is displaced by a distance S um So there's some things that we're going to need. First of all, we are going to uh set up the force net net force on that little charge by finding the total electric field acting audit. Um so those two charges as the charge gets displaced in the center, uh we'll have different magnitudes with the one being a little bit weaker and e to being a little bit stronger. And so uh we can use the magnitudes of those to calculate the X. Component of the force on that little charge. And that exponents, that X component will govern the mass times the acceleration of the central charge which has masked M Okay, so that is our goal is to find that net force. Of course we will need columns law. So the magnitude of the electric field from a point charge is K. The electrical constant times the queue over R squared. I'm also going to write down hooks law. We won't work with the differential equation we get, but just by analogy will look at hook's law, which says if you have a force that is proportional to the displacements. Um you basically wind up with simple harmonic motion with angular frequency equal to the spring constant Ks over em. And that's also related to the period of the oscillation. Um hooks laws, the force law, The simple harmonic motion comes about from solving Newton's second law for that type of horse. Okay, so we're ready to put in uh once we we figure the um force out and that indeed undergo simple harmonic motion. Well, look at a simple numeric example. Okay, so let's put in the two electric fields. Uh the electric field # one is a little bit weaker. Oh yeah, let's put all the constants out front. So the separation between the little Q and the big Q to the left gets a little bit bigger and it gets a little bit smaller to the right the way I've drawn it. Yeah. Okay. Now to simplify this, what we're going to be doing is using the binomial approximation, which is a very handy tool to linear eyes things. What this says is if you have the binomial one plus delta and you raise to the power, uh that could be there plus or minus delta. Um That is approximately one plus and delta. And it works for negative exponents and or positive exponents. And but you must have delta Much less than one. Okay, so it's an approximation and in our expressions we can take something like one plus D. To the S plus S squared in the denominator and approximate it in our nice by no meal form. Okay, because we have with, once we take the D squared out, we see that we have one plus delta raised to the minus second power. So that is an approximation for the first term. And we can do a very similar thing for our second term. And then if we add those two terms together actually not add them but subtract them. We see that we are left with -4 S over D cubed. Yeah. Okay. So the constant term disappears and we are left with twice -2 S over the cubed. So we can now after that approximation it's a reminder that is an approximate thing. You sing the binomial approximation. We can now write our X. Component of force as cute little cute big Q electrical coefficient And then -4 s over D cubed. And yeah, I can rearrange that. So it looks more hooks law like with a lot of constant numbers out in front. I'll put those all in parentheses for a little Q. Big Q electrical constant divided by the cube times the displacement. S. So yes, our force law that we have found from those two electric fields looks like hooks law. And the quantity and parentheses, we are going to call it our spring stiffness or spring constant. And now we know that the acceleration, the second derivative of S. Is proportional to S with a negative sign. And so the charge will oscillate. Little Q will oscillate. And after using that in our period relationship what to pi over the period is spring constant over m square root. We can rearrange that and we come up with formula for the period. It is two pi am over the spring constant where the spring constant. I probably don't want to plug all that in. It'll look a little bit messy. Um But we do know what that constant is and here's where things get a little bit of awkward. There's two K's. One is the electrical Uh huh, fundamental constant of electricity. We'll get to that. The other is a particular spring constant. So yeah, let's do an evaluation. Let's try to find the oscillating period if we somehow have a sodium ion in a crystal Stuck between two other fixed sodium ions. Okay, so we're going to make all the charges equal to each other and equal to the fundamental electrical E um We are going to need well, we we have the electrical K Is 8.99 Times 10 to the 9th. In a Newton's Times meter squared over Coolum squared. Um and we are going to need the d spacing, which we are just going to approximate as three Times 10 to the -10 m. That is a good size of an atom May not be perfect, but we'll go with it. Um and then the mass of a sodium atom is 22 atomic mass units. And I am using 1.67 Times 10 to the -27 kg for the atomic mass unit. Okay. And I'm not going to show all the calculations but that shows all the numbers being used. I will show um the intermediate spring constant. Mhm. Uh is for oh yeah And winding up with 34.1. Uh Lieutenant's perimeter students per meter when I use all the values uh in the equation there, check that out. And so my period then is quickly found from the mass divided by the K spring constant. And if I have used S. I. Units, uh the period should wind up to be in seconds and I will go ahead and convert it into PICO seconds. So I'm getting a fraction of a PICO second. So that gives you an idea of how rapidly an atom will vibrate stuck in a crystal. Okay.

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