(a) A set of point charges is held in place at the vertices...

(a) A set of point charges is held in place at the vertices of an equilateral triangle of side $10.0 \mathrm{cm},$ as shown in Figure 18.38$($ a). Find the maximum amount of total kinetic energy that will be produced when the charges are duced when the charges are released from rest in the friction less vold of outer space. (b) If the charges at the vertices of the right triangle in Figure 18.38$(\mathrm{b})$ are released, how much total kinetic energy will they gain? When will this maximum kinetic energy be achieved, just following the release of the charges or after a very long time?

Key Concepts

Electric Potential Energy

This concept refers to the energy stored in a system of charges due to their mutual positions. In the context of point charges, the electric potential energy is calculated using Coulomb's law and represents the work required to assemble the configuration from infinity. When the charges are released, the decrease in the system's potential energy is converted into kinetic energy, making it a key factor in determining the maximum kinetic energy obtained by the charges.

Conservation of Energy

The principle of conservation of energy is central to these problems, stating that the total energy of a closed system remains constant. In the case of charges being released, the initial electric potential energy is completely transformed into kinetic energy (assuming no losses to other forms of energy). Understanding this conversion allows one to calculate the maximum kinetic energy the charges will acquire based on the initial potential energy of the system.

Coulomb's Law

Coulomb's law governs the interactions between point charges and provides the mathematical framework for both the force between the charges and the potential energy stored in the system. It is essential for calculating the pairwise electric potential energies in the configurations, which then are summed to yield the total potential energy available for conversion into kinetic energy.

Configuration Geometry

The spatial arrangement of the charges, whether in an equilateral triangle or a right triangle, directly affects the distances between the charges and thus the magnitude of the potential energy as described by Coulomb's law. The geometry is integral to solving the problem because it determines the pairwise distances, which in turn dictate the relative potential energies and the total kinetic energy that can be generated when the charges are released.

Transcript

00:01 This question gives us two configurations.

00:03 One is in an equilateral triangle, and the other one is in this right angle kind of configuration.

00:12 And what we are asked is that if we convert all the energy in this configuration into kinetic energy, what is the maximum kinetic energy we can get out of each configuration? so this question can be accomplished if we think about the conservation of energy.

00:33 So right now is when everything is fixed in their configuration, there is no kinetic energy because nothing is moving.

00:40 But at the same time, each configuration has their own huge potential energy.

00:46 And when the charge are let loose, what will happen is that the charge will move apart from each other because they are all of the same signs.

00:57 So the electric force they have on each other repels.

01:00 Them.

01:01 So they move in different directions and they will accelerate.

01:05 And because of that, as they move apart, the potential energy decreases and the potential energy is converted to the kinetic energy.

01:15 When they're infinitely far, that is when their potential energy is zero, that is when all the potential energy will be converted into kinetic energy.

01:25 So in order to find the maximum kinetic energy that is to be produced, we find, the potential energy storing this configuration because that's the total amount of energy and that's the maximum amount that could be converted to kinetic energy.

01:41 And for potential energy, we already know that potential energy u equals kq times q prime over r.

01:51 This is for a single charge q in q prime.

01:54 We have multiple charges, so the actual u will be a sum of these.

01:59 So it will be k over q1, q2 over r12 plus q2 2 over r2 3 over r2 3 plus q3 q1 over r31.

02:16 So it is the first configuration we can see is equilateral, so the distance between each of them are the same.

02:22 It's all 10 cm is what is given in the quastricht.

02:27 So we just plug in all the numbers.

02:31 We have 8 .99 times 10 to the negative 9 over.

02:40 So it's 60 .0 .1, i'll put some at the very beginning times.

02:48 Okay, so q1, q2, q's 3 are all of different charges.

02:53 So i have 4 times 10 to the negative 6 times 2 times 10 to the negative 6.

03:00 That is the left two charges.

03:06 And then i have two times 10 to the negative six times three times 10 to the negative.

03:12 That's the bottom two...

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