Review problem. Two identical particles, each having charge +q, are fixed in space and separated

Review problem. Two identical particles, each having charge...

Key Concepts

Taylor Series Expansion

A Taylor series expansion is used to approximate the behavior of functions near a given point by considering terms up to a certain order. In this problem, expanding the expression for the net electrostatic force in powers of the small displacement from the equilibrium position helps simplify the analysis and identify the linear term responsible for the simple harmonic motion.

Small Oscillation Approximation

The small oscillation approximation involves considering only the first non-zero term in the Taylor series expansion of the potential or force when the displacement is small. This approximation simplifies the force to a form that is linear in displacement, which is a key characteristic of simple harmonic motion. It enables the derivation of the effective spring constant and the oscillation period for the system.

Energy Conservation

Conservation of energy is a principle that equates the total mechanical energy (potential plus kinetic) at different points in the motion. In part (b) of the problem, energy conservation is used to relate the change in potential energy of the moving charge as it moves from a small distance away to the midpoint, to the kinetic energy it acquires, allowing for the calculation of its speed at the midpoint.

Coulomb's Law

Coulomb's law is fundamental for calculating the electrostatic force between charged particles. It states that the force between two point charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. This principle allows us to determine the forces acting on the moving charge from the two fixed charges.

Simple Harmonic Motion (SHM)

Simple harmonic motion describes oscillatory motion where the restoring force is directly proportional to the displacement from equilibrium. In this problem, when the moving charge is slightly displaced from the midpoint, the net electrostatic force behaves like a linear restoring force, enabling one to treat the motion as SHM. This approximation is valid when the displacement is small compared to the separation of the fixed charges.

Transcript

00:02 Okay, let's say we have two charges, both are positive, and they're separated by a distance d.

00:07 So you can see the left and right charges on the diagram here.

00:10 Those are the charges that are separated.

00:12 And let's say we have a third charge, but this charge is negative.

00:16 This charge has a capital q, and it's below these charges.

00:19 And it's actually a distance of x right below the midpoint that separates the two charges.

00:26 So, like, as you can see here, that black dot in the middle of the red line, that is the the midpoint.

00:32 So what we want to do is we want to prove that this negative charge on the bottom is going to travel in simple harmonic motion.

00:39 That means it's going to cycle up and down in between these two charges.

00:45 Because if you think about simple harmonic motion, and i'll just call that shm, just to make it a little easier, that's kind of like a spring where you're going up or down.

00:57 And two important things about simple harmonic motion is that we know that for simple harmonic motion, the acceleration is going to be proportional to the displacement.

01:10 And the proportionality constant is going to be omega.

01:15 And omega is what's called the angular velocity.

01:20 But all we know is that that is a constant.

01:23 So if we can find an equation with the acceleration and if it has a x in it, then we can prove that there is simple harmonic motion.

01:33 Another part of simple harmonic motion is the velocity is also proportional to just omega -x.

01:41 So we can prove either one of these.

01:43 We know we're in harmonic motion.

01:46 Now, i actually want to use acceleration.

01:48 The reason i want to use that is because i know i got a bunch of charges here.

01:52 So i know that they all feel a force via kulam's law.

02:00 And kulam's law states that the charge between any two charges, sorry, that the force between any two charges is going to be k times the first charge, times the second charge, over the distance between the charges squared, which we'll call r squared.

02:20 So we know that we can use this to find the total charges, and then we can also use newton's second equation.

02:27 Equation, f equals m .a.

02:31 And we can rewrite that in terms of acceleration, say a equals f over m.

02:37 So that means if we can find our electrostatic force through kulom's law and then divide it by m, that's going to give us the acceleration.

02:47 And then we can check to see if it's proportional to the displacement x.

02:51 All right.

02:52 So let's get started.

02:54 So first, i want to just look at the left side because it's always easier when you're only looking at like a two charges and a once.

03:02 So i want to start just looking at this side.

03:07 So first i'm going to draw my distance between the two charges because remember, i need that r squared.

03:14 So i'm going to draw r here.

03:17 And then i also know this is going to form a right triangle.

03:21 Right.

03:21 So i have r is my hypotenuse.

03:23 I have my leg, got x and because i know that this top leg is half of the distance d because i remember this negative chart is right below the midpoint this leg is d over two so that means i can use the pythagrin theorem and i can write this as x squared plus d over two squared equals r squared right and the cool thing there is now i can just replace everything that i have here with like what i know.

04:04 So i have k, i have my positive q.

04:09 I have my negative q as well.

04:11 And like that's a big q.

04:15 And then i can just put r squared, which i know again is x squared plus d over 2 squared.

04:25 All right.

04:26 So if i pull that negative out, i'm going to.

04:30 Have kqq over x squared plus d over two squared now remember this is just for the left side here and that force is going to be directed toward each other because it's negative that means that's going to be an attractive force and that it should make sense because we have a positive and a negative here now if we look at the other two we can see that these are going to have the exact same force because we know that this x is the same and we know this is also d over two so that means that we know that this r is going to be the the exact same so if we say that this is the force from the left charge then we can also say that we know that the force from the right charge is going to be the exact same which means that our total force is just going to be the sum of both of these, which means i can just take one and multiply it by two.

05:45 So this is going to be the total force between these three charges.

05:54 Okay.

05:55 So now, actually, i'm going to re -jard this so it's not as messy.

06:00 So now i'm just going to draw one pair of charges because remember, we only need one pair because it's like a symmetrical.

06:09 So we mentioned that this is going to be an attractive form.

06:11 Force.

06:13 Now, we need to break these into components.

06:19 So if we break these into components, we can see that we're going to have a, this is going to be our x component of fl.

06:29 This is going to be our y component.

06:31 Right.

06:33 So we know that if we think about our other charge, because remember we have another one with that same force, we're going to have those same components, except now they're coming from the right charge.

06:45 So f, x, fry.

06:51 So now we can see that these two have to cancel out because they're both equivalent magnitudes pointing in the opposite direction.

07:00 So in the same way you have negative 2 plus 2 2 equals 0, both of these have to be equal to like 0.

07:08 So what that means is that the total charge on here is really just the y component, right? and we know that the y component of a vector, again, if we draw our triangles back in, if we say that this is our theta, then we know that our y component is just going to be f -l -y times sign of that angle, because we can use trigonometry for that.

07:35 So what that means is that our real total f does not include x.

07:42 So that means that we want to have our formula for f, but then we want to multiply it times sign of the angle.

07:55 And if we look at what sign means, we have opposite.

08:00 And remember, this side is x.

08:02 So that will be our opposite.

08:04 And we have our hypotenuse.

08:06 And this side is r...