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Integrated Concepts (a) In Figure 18.59 , four equal charges $q$ lie on the corners of a square. A fitth charge $Q$ is on a mass $m$ directly above the center of the square, at a height equal to the length $d$ of one side of the square. Determine the magnitude of $q$ in terms of $Q, m,$ and $d,$ if the Coulomb force is to equal theweight of $m .(b)$ is this equilibrium stable or unstable? Discuss.

a)$q=\sqrt{\frac{3}{2}} \frac{3 m g d^{2}}{8 k Q}$b)Equilibrium is unstable.

Physics 102 Electricity and Magnetism

Chapter 18

Electric Charge and Electric Field

University of Michigan - Ann Arbor

University of Washington

Simon Fraser University

Lectures

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okay. In this problem, we have a square of charges in a what side B. We're s defying First, the total electrostatic energy of this configuration from there were asked if we were to put a charge at the origin. What would the total electrostatic energy be due to the 55 point configuration? And then we're asked if we were to push that point at the center of the square, offset it a bit? Would this be a stable equilibrium, or would it be unstable? It's a start. We can begin constructing the total energy recall that the energy electrostatic energy between point charges Q Times V, which would, for the potential of a point charge apparel wise electrostatic energy would be simply Q one Q two over four by absolutely not are between them. So we're gonna do this for every single pair combination of our on our square. So our total, uh, electrostatic energy will be one of her four pipes or not. Do you want you to over our 12 plus you want Q three are 13 was Q one Q four R 14 Now we have to consider some cross terms. Cue to cue. Three are 23 Cue to cue for RU four and lastly, you drink If you're our 34 Okay, so fortunately, the firm for the first a couple of terms the same distances and also these are all the same charge that's that makes things useful. We can pull a Q, score it out and we got one over B. It's one of a route to be That's one of her. B. It's one of me. That's what we're to be whatsoever. Be where the one overby distances are between sides. And then we have cross terms of the one over route to be just a bit of geometry, and this all simplifies down, too. You squared over four pi Upsilon, not four plus 320 and we can pull that b out go. And that's the electrostatic energy due to the four charges in the square. Now, if we add a point to the origin branch are Georgian, it's electrostatic energy, you fifth, but simply be four would be that we have to consider contributions of the four other charges, so we'd have to add we have to take into account each of these distances we're gonna be rich to be each. So we would have keep guard over four by excellent Not be. That's four Ruutu. This is what we would have to add the energy dad to place it there because it's gonna have interactions with four other charges. Okay, Nothing about if we move this charge in any direction, what would happen? Well, if we moved it so it was slightly closer to one charge would be pushed back because on these charges have the same sign. It's a repulsive force. So there was any closer to any charges should be pushed back with greater force which would expect to leave. Put your back to its original position right in the center. This is also the minimum energy state. When the party is at the very middle, the total electrostatic energy is at a minimum. This is where the system is the most relaxed. So for these reasons, we call this a stable equilibrium because if you try not try toe destabilize it, it will stabilize itself. Stable equilibrium and that's it.

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