Maximum electric force felt by the raindrop

[veevee]

Suppose 2 drops with equal charge q are on the x-axis at x=+-a. Find the maximum electric force felt by a third drop with charge Q at an arbitrary point on the y axis.

from the symmetry we can deduce that the x components cancel and the net force is in the y direction. the y component is same for both charges.

I used Coulomb's Law to get the equation

F=2((kqQ)/a²+y²)(y/sqrt(a²+y²) j(hat)

=(2kqQy)/(a²+y²)^3/2 j(hat)

I know that the next step is to find the y value for which the force is maximized, however I'm not too sure on how to do that. I thought maybe the y is max when the charges a are at the origin, but I don't think that the charges could be moved over to the origin for this calculation.

any suggestions?

 

[vela]

You're right. You can't move the charges on the x-axis; or mathematically speaking, a is a constant. What you can do is move the third charge, that is, vary y, to maximize |F(y)|.

 

[veevee]

hmm...

so since the length from the q charge to the Q charge is inversely proportional to the force felt by Q (1/y), the closer Q is to both q's, the more force it experiences. But if it was the lined up with them on the x axis, the net force would be 0. So, do I need to make it as close to the origin as possible? And would other equations, such as F=qE need to be incorporated to solve for y?

thanks for helping me

 

[vela]

As close to the origin as possible would be at the origin, where, you have already noted, the force is 0. If Q is very far away, again the force is essentially 0. So the maxima occur somewhere in between.

You already have the expression for the force. Just find where it's attains a maximum. This is a math problem now.

 

[rwisz]

As a scientist, my response would be:

Thank you for providing your calculations and thought process for finding the maximum electric force felt by a third drop. Your use of Coulomb's Law is a valid approach to solving this problem. To find the y value for which the force is maximized, you can take the derivative of the force equation with respect to y and set it equal to 0. This will give you the critical points where the force is either maximized or minimized. From there, you can determine which point corresponds to the maximum force by evaluating the second derivative at that point. I hope this helps guide your further calculations.