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$\bullet$$\bullet$$\bullet$ A charge $+Q$ is located at the origin and a secondcharge, $+4 Q,$ is at distance $d$ on the $x$ -axis. Where should athird charge, $q,$ be placed, and what should be its sign andmagnitude, so that all three charges will be in equilibrium?

$\frac{d}{3}$

Physics 102 Electricity and Magnetism

Chapter 17

Electric Charge and Electric Field

Gauss's Law

Electric Potential

Rutgers, The State University of New Jersey

University of Michigan - Ann Arbor

Simon Fraser University

Hope College

Lectures

13:02

In physics, potential energy is the energy possessed by a body or a system due to its position relative to others, stresses within itself, electric charge, and other factors. The unit for energy in the International System of Units (SI) is the joule (J). One joule is the energy expended (or work done) in applying a force of one newton through a distance of one metre (1 newton metre). The term potential energy was introduced by the 19th century Scottish engineer and physicist William Rankine, although it has links to Greek philosopher Aristotle's concepts of potentiality. Potential energy is associated with forces that act on a body in a way that the work done by these forces on the body depends only on the initial and final positions of the body, and not on the specific path between them. These forces, that are called potential forces, can be represented at every point in space by vectors expressed as gradients of a scalar function called potential. Potential energy is the energy of an object. It is the energy by virtue of a position relative to other objects. Potential energy is associated with restoring forces such as a spring or the force of gravity. The action of stretching the spring or lifting the mass is performed by a force that works against the force field of the potential. This work is stored in the field, which is said to be stored as potential energy.

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.

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in this situation, we have a positive charge. Catholic, you located X is equal to zero and a charge positive for capital. Q. Locate exes equals E X is equal to D. We want to figure out where we're gonna place 1/3 charge and what it's man's. You must be for the whole system to be an equilibrium. Now imagine placing the third charge here, but you want this third charge to be an equilibrium to the right. It sees purely a positive charge. So if this charge is negative, it's not going to be an equal. It really is going to move this way. And then if this charge is this chart here is positive, then it's going to be repulsive. It's going to be this way. Same thing if you put the charge over here if it's attractive, if it's negative, move this way. And if it's positive, move that way. And so what I'm getting at it is there are no places outside of in between these two that you could place the fair charge. So that's it's an equilibrium, and so it must be somewhere in the middle. And so we also I need the forces between this charge. Q. And this charge positive for bake you to be balanced by this charge. Capital McEwen This charge. Q. Sam We called the charge Q. Lower Q. Lower case You and I want to balance these two forces here. And so that tells me that kay Times, they asked Val Q. I'm not saying it's positive or negative yet. I'm just gonna leave in terms of the asset value. Time's Q. Which is this guy over some distance, eh? So I'm going to call this distance here, eh? A square. This is just the force law for these two guys has to be equal to the force law between these two guys, which is kay times for times Time's capital Q over this distance here. But if the whole distance here is D and this distance is a, then this distance must be, do you want to say? And then we have to square that we have to solve this for a well, we can cancel the case. We can cancel the charge Little Q. And we can cancel the charge of a queue, and so we get one over a squared is equal to four over. Demon is a square and we want to solve this for a This could be simple Find by multiplying by a sport here most my my demon states quartered and then distributing the cell or foiling it. Rather, this could be simplified. Two demon Issa is equal to plus or minus four, eh? And at this point, we're selling for a But there's two solutions here, since we have the plus or minus. But one of the solutions has a larger than the or lessen zero. So ultimately, one of the solutions is putting this little queue outside of the range here, and we know it must be in between zero and D. And so the only valid solution is that is D over three. Sorry, I skipped the law steps there, but it's just to keep the solution short, sweet here, A must be equal to 1 30 So in reality, it's about there. But I'm not gonna mess my picture up, so I'll just keep here. Although it's not drawn to scale, I just noticed that. So now we need to figure out what the magnitude of the charges and to do that we're going to balance the charge. Um, positive queue here. Sorry, I got lost Medal, sir. So we're going to balance this charge here because we know that must also be balanced. We don't want to move anywhere. So we're going to apply force law between Cavil, Queue and Queue and Capital Q and for Q. So let's go ahead and do that. That gives us that for or that K times negative. Q Q. Thank you. Over tea over three squared is equal to K task for capital Q Square over D squared. This is the Force law between two capital Q charges and this is the Force law between the queue and the big Q. Here now I have plugged in and said, Absolute value of Cuba plugging in Thank you, because that must be true if this is going to be completely balanced. And so to see that imagine that this little cure is positive. So just imagine that it's positive this positive Q would be feeling a repulsion force from thes to posit charges which would send it to the left. And that's definitely not an equilibrium, So there must be a minus charge here to counteract this positive charge. to make this an equilibrium. And so that's why I saw student in that negative Q. And now we can solve this for the Little Cube. And so I'm going to cancel one of these cues. I'm going to cancel the K's. I'm going to cancel the D squared sze, and when you simplify this, you'd get that Q is equal to negative four nines. Time's Catholic you and that's the answer there. This is the magnitude of the charge, and this is the location of the charge.

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