Find the net force on charge

[aaaa202]

Homework Statement

13 equal charges are placed in the corners of a regular 13-sided polygon. What is the force on a test chrage at the center?

Homework Equations

principle of superposition
symmetry

The Attempt at a Solution

I am not sure if the net force is 0 or not. Had the polygon been 12-sided the symmetry would be obvious, but I'm not sure in this case. Can anyone explain if the net force is 0 or not?

 

[tiny-tim]

hi aaaa202!

13 equal charges are placed in the corners of a regular 13-sided polygon. What is the force on a test chrage at the center?

try it first for an equilateral triangle (3 equal charges) …

is that 0 ?​

then use the same method for 13 (you'll need a formula for summing a trig series)

 

[aaaa202]

Did the calculations for a triangle as you said, where i fixed the coordinates such that one of the charges sat on the y-axis. From there it was easy to see that Fres = 0, and I think that this must hold for any rotation of the coordinate system. After all I think the length of a vector is more or less defined to be preserved on a rotation in the euclidean coordinate system. However, I'm not sure, so can you confirm this? :/
And can you perhaps come up with a symmetry argument that makes it easy to see, that rotation of the coordinate system should not change the lFresl? :)

 

[tiny-tim]

Did the calculations for a triangle as you said, where i fixed the coordinates such that one of the charges sat on the y-axis. From there it was easy to see that Fres = 0

yes!

presumably you did cos0 + cos2π/3 + cos4π/3 = 1 - 1/2 - 1/2 = 0 ?

ok you now need to prove cos0 + cos2π/13 + cos4π/13 + … cos24π/13 = 0

 

[asteriatic]

I would approach this problem by using the principle of superposition, which states that the total force on a charge is the vector sum of all the individual forces acting on it. In this case, we have 13 equal charges placed in the corners of a regular 13-sided polygon.

Since the charges are equal and symmetrically placed, we can assume that the forces acting on the test charge at the center will also be equal and opposite due to the symmetry of the polygon. This means that the net force on the test charge will be 0, as the individual forces will cancel each other out.

To verify this, we can use the formula for the force between two charges, which is given by Coulomb's law: F = kqQ/r^2, where k is the Coulomb's constant, q and Q are the charges, and r is the distance between them. Since the charges in this problem are equal, we can simplify the formula to F = kq^2/r^2.

Using this formula, we can calculate the force between the test charge at the center and each of the 13 equal charges at the corners. Since the charges are equal and the distance between them is the same, the forces will also be equal in magnitude but opposite in direction. Therefore, when we add up all the forces using the principle of superposition, the net force will be 0.

In conclusion, the net force on the test charge at the center of a regular 13-sided polygon with 13 equal charges at the corners is 0, due to the symmetry of the charges and the principle of superposition.