Review. Two identical particles, each having charge +q, are...

Key Concepts

Taylor Series Approximation

The Taylor series approximation is a mathematical method used to expand complex functions into simpler polynomial forms around a specific point, often an equilibrium point. In the context of physics, this technique allows for the approximation of forces or potential energies to leading order terms, which is essential in deriving the equations of motion for systems undergoing small oscillations.

Small Oscillation Approximation

The small oscillation approximation involves linearizing a system’s restoring force near its equilibrium position by assuming that displacements are sufficiently small. Under this assumption, nonlinear terms can be neglected, resulting in a harmonic oscillator model where the motion is described by simple harmonic motion.

Conservation of Energy in Electrostatics

In electrostatic systems, the principle of conservation of energy holds, where the total mechanical energy (the sum of kinetic and potential energies) remains constant. This concept is utilized to relate changes in potential energy due to the electric forces between charges to the resulting kinetic energy, which can be used to determine speeds and dynamics in the system.

Simple Harmonic Motion (SHM)

Simple harmonic motion is a type of periodic oscillatory motion where the restoring force is directly proportional to the negative of the displacement from equilibrium. This concept is central in physics for modeling systems that, when displaced slightly from equilibrium, experience forces that lead to oscillations with a constant period.

Coulomb's Law

Coulomb's law describes the force between two point charges, stating that the magnitude of the electric force is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. It is fundamental for analyzing interactions in electrostatic problems and determining the forces that act on charged particles.

Transcript

00:01 Okay, so this is the, uh, rafay drew for his question.

00:05 So there's a mega charge along.

00:07 Asked access and father charge along.

00:10 Why access on each side and distance? uh, between the positive charge, each positron to the original points d over two.

00:19 And isn't it? they were charged to the original point is acts.

00:23 Okay, so the force acting on the negative charge too.

00:28 Show me two times the force on the house of charge to the native charge ties science data.

00:41 Okay.

00:42 It was their angle here.

00:43 Okay, inside today's along the x axis and ah, forces along the y axis wishes the forces on a lot of wine components is gone because you cancel out eventually, okay? because it looks something i did.

01:00 And eventually his forces cancel out.

01:07 So therefore, we'll have two times, you know, on the force from the positive charge.

01:14 Don't use i should be que se que okay.

01:20 Skewed has uppercase que over, um be over to square plus ax squared.

01:35 Ok, because the distance between the each positive charge to a neighbor charge is the dashed line here.

01:42 Okay, which is x square plus 3/2 square, which is they are score here.

01:49 Okay.

01:51 And then times, scient, ada, we know scientific.

01:53 The is equal to x over and then for the here.

02:01 So you know what? until this you go to acts, overs square root de over two square plus x square.

02:17 Okay, because this is the distance between the positive part in it and the negative charge.

02:23 Ok, as things we know that acts, it's a smaller than the air force.

02:32 Well, if oh so d over two square plus x were or just be approximately equal to de over two square day.

02:47 Therefore, you totally larger force on the negative charge.

02:51 Q is equal to two times que que que over do you over to square times x over square ooh de over to and square, which will give us to thank you que t over de over to to the power of three times acts.

03:38 And these expression here it's the expression for the single i'm in motion.

03:51 Okay.

03:57 And if the question be was asking us to determine a period of emotion what we know since the force you see, it was who two.

04:14 Okay.

04:16 Que que over do you over to over 3 10 acts.

04:25 I was something the force is equal to mass times acceleration, which is equal to justin here.

04:36 E don't know that acceleration is equal to anger.

04:44 Philosophy.

04:44 Square times acts therefore half m times omega square test acts.

04:54 You see what's who.

04:56 Two can q q over do you over to two of our three and intense acts.

05:04 As you can tell, x x can became so well careful would have m omega square.

05:12 It's people too.

05:15 Tu que que que over do you over to to the power three.

05:25 And we also know that oh my god, which is the end of last year is equal to one.

05:31 Revolution was able to pi over the time here, so have omega square is equal to two.

05:43 Okay, que que over m times do you over to to the power three and i will have two pi over tea square.

06:06 You see what to two.

06:09 Okay, joke you over and honesty over to to the power three who have well, hi square over t square is equal to two.

06:26 Okay.

06:28 Que que over and dynasty over to to the power three.

06:33 And we also know that okay, which is a cool on constant can be equal to 1/4 pi has absolutely not absolutely zero.

06:44 Okay.

06:45 And i think unplugging this back until the equation here and we'll have well by square over t square is equal to two over or by absolutely ties que que over and do you over to the power three and then, as you can tell to cancel here, then we can move one over to buy tens epson on zero to a website, and we have well by square times to buy absolutely zero over t square is equal to you.

07:40 Kill over m times c over to to about three rome.

07:51 We'll have a t square.

07:54 You see, you go to a pie cube...

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