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Q)Given that q1+q2=q. For what ratio q1/q will the force between q1 and q2 be maximum?

Hello Ishaan,

We have to assume that the distance between the 2 charges remains constant. So the force varies as the product of q1 and q2. So, for what value of q1 does the following function have a maximum:
f(q1) = q1*q2
Since q2 = q - q1, the function can be rewritten
f(q1) = q1*(q - q1) = q*q1 - q1^2

Setting the first derivative equal to zero will tell us when the slope of the function is zero and therefore either a maximum or minimum. Since q is a constant, the first derivative is
f'(q1) = q - 2q1
Setting that equal to zero we find that q1 = q/2.

We can find if that is a minimum or maximum from looking at the 2nd derivative.
f''(q1) = -2
Since the 2nd derivative is negative, f(q1) is maximum when q1 = q/2.

I hope this helps,
Steve
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Physics

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Steve Johnson

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I would be delighted to help with questions up through the first year of college Physics. Particularly Electricity, Electronics and Newtonian Mechanics (motion, acceleration etc.). I decline questions on relativity and Atomic Physics. I also could discuss the Space Shuttle and space flight in general.

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I have a BS in Physics and an MS in Electrical Engineering. I am retired now. My professional career was in Electrical Engineering with considerable time spent working with accelerometers, gyroscopes and flight dynamics (Physics related topics) while working on the Space Shuttle. I gave formal classroom lessons to technical co-workers periodically over a several year period.

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BS Physics, North Dakota State University
MS Electrical Engineering, North Dakota State University