What Value of q Maximizes the Repulsive Force Between Two Charges?

[aaronmilk3]

Derive your results algebraically… No graphical results.
An excess charge of +Q is distributed into two unequal parts, one having charge +q .
Assume that these two new charges are now separated by a non-zero distance. What value of q maximizes the repulsive electric force that one of these new charges exerts on the other?


I got so far that q and the other one is charged (Q-q) due to the repulsive force. I know I would need to take the derivative but this is where I got lost. Any help would be great! Thank you in advance!

 

[collinsmark]

I got so far that q and the other one is charged (Q-q) due to the repulsive force. I know I would need to take the derivative but this is where I got lost. Any help would be great! Thank you in advance!

You're on the right track. As a matter of fact, it sounds to me like you almost have it already!

So go ahead and set up your force equation, with one charge q and the other charge (Q - q), separated by some distance.

To maximize a function, take the derivative with respect to whatever variable you wish to vary, and then set the result equal to zero. Solve for the variable (algebra). That's the value of the variable that maximizes or minimizes the function.*

*The answer can give local maxima or local minima. So if you get multiple values (such as in a quadratic), you might wish to do some sanity checking to figure out what each value is. In this particular problem though, you end up with a single maximum (no minimum) so the need for sanity checking doesn't apply here.

The idea behind the procedure is this: When a smooth function rises to the top of its [possibly local] maximum or bottom of its [possibly local] minimum, the slope of the line is zero at this point. So the above procedure is merely finding where the slope of the function equals zero, as the variable changes.

 

[supratim1]

yes, right, take the derivative with respect to q (that is d/dq), and solve as above.

you should get Q = q/2