Deriving equation for electrostatic force.

[Checkfate]

Homework Statement

Basically I am given an image showing a test charge tied to a string where there is the ball is stationary. I need to use the fact that there is no net force on the ball to derive an equation for the electrostatic force in terms of g, the angle, and the mass of the ball. Unfortunately I am stuck. :(

Homework Equations

The Attempt at a Solution

Well I know that the vertical portion of $$F_{T}$$ is equal in magnitude to mg, and the horizontal portion of $$F_{T}$$ is equal in magnitude to the electromagnetic force. But I don't see how to tie it together to derive an equation.

 

[neutrino]

How about trying to eliminate F_T from the equations.

 

[Checkfate]

Ok sure let me try ;-).

If I am speaking about magnitudes only,

$$F_{T}^{2}=F_{g}^{2}+F_{e}^{2}$$

Or

$$F_{e}^{2}=F_{g}^{2}-F_{T}^{2}$$

But $$F_{T}=\frac{F_{E}}{Sin\Theta}$$

So

$$\frac{F_{e}^{2}}{(Sin\Theta)^{2}}=F_{e}^{2}+F_{g}^{2}$$

$$F_{e}^{2}=F_{e}^{2}(Sin\Theta)^{2}+F_{g}^{2}(Sin\Theta)^{2}$$

moving the fe^2(sin(theta))^2 to the left side then factoring out 1-sin(theta)^2 from the left side and converting it to cos(theta)^2 I get:

$$F_{e}=\sqrt{(tan\Theta)^{2}m^{2}g^{2}}$$

Does it look right?

 

[neutrino]

Yes that's right. But a simpler way, actually the simplest, would have been to divide the expression for F_e by that of F_g.