Question

Four objects, each with charge +q, are held fixed on a square with sides of length d, as shown in Figure 18.66. Objects X and Z are at the midpoints of the sides of the square. The electrostatic force exerted by object W on object X is F.
What is the magnitude of force exerted by object W on Z?

- F/7
- F/5
- F/3
- F/2

Question Image

Final Answer

(b)

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This is College Physics Answers with Shaun Dychko.
We're given a square with side length

*d*, and four charges on it of equal magnitude,*q*. Charges*x*and*z*are at the mid-point of their respective sides. So the force between*w*and*x*we're told is*F*. We'll call the distance between them <>r x. The question is what is the force on charge*z*? The separation between*w*and*z*is*r*subscript z. So the force on*z*is going to be*k q*squared over*r z*squared and the force on*x*which is just called*F*is going to be*k q*squared over*r x*squared. Now the question is how do we relate*r x*and*r z*? If we do that we'll be able to relate these two forces. So*r x*is*d*over two because it's at the mid-point of the side and*r z*is going to be the hypotenuse of this triangle here, and this triangle has a side length of*d*on one side and*d*over two on the other side. So*r z*is going to be the square root of the sum of the squares of the legs of this right triangle. So that's square root of*d*squared plus*d*over two squared. That's*d*squared plus*d*squared over four. This is four over four*d*squared you could say, and that makes a total of five over four*d*squared under the square root sign. Then we'll do the square rooting. So we get square root five over four times*d*which is root five over two, times*d*.*d*over two is*r x*so we can substitute*r x*in place of*d*over two. So*r z*is root five times*r x*. Now that means we can replace*r z*with this. So we've done that here. So*F z*is*k q*squared over*r z*which we've replaced with root five times*r x*squared and that makes one fifth*k q*squared over*r x*squared and*k q*squared over*r x*squared is*F*. So*F z*is one fifth*F*. So the answer is B.