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Problem:

A uniformly charged circular arc with linear charge density $m$ subtends angle $\theta$, find the net force acting on a charge placed at its center if total charge of the arc is $Q$ and the charge at center is $q$.

Here, we take the line of symmetry for the arc and integrate the components of force acting along it i.e: $$2\dfrac{kqm}r\int_{0}^{\theta /2}(\cos x)$$ We get: $$2kqm\dfrac{\sin(\theta / 2)}r$$

If we integrate from $\theta/2$ to $0$, we get negative value ($k = 9 \times 10^9$, $r$ = radius of arc)

I wanted to know, why is the sign changing when we add the same thing in different ways. Is there a rule to know how to decide the limits or its just the faulty mechanism or there is actual change in direction of force.

I was thinking that it actually rotates the axes about y axis, where positive z axis is decided by using right thumb rule hence as the force acts on x axis, the sign gets flipped. Just a speculation.

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    $\begingroup$ Hello and welcome to Physics SE! When posting, please make sure to format your question with the correct punctuation, spellings, and capitalization. Use LaTeX to format symbols and math. $\endgroup$ – Yejus Jul 30 '20 at 16:43
  • $\begingroup$ Thank you for the edit, sorry i am new to the platform, i will be more careful from the next time $\endgroup$ – Harshavardhan Hajeri Jul 30 '20 at 17:00
  • $\begingroup$ This is just a pure mathematics question concerning integrals in calculus, when you change the direction you have to add another negative sign is all. $\endgroup$ – Triatticus Jul 30 '20 at 18:05
  • $\begingroup$ What Triatticus said. If you like, this is because even though you are still just adding the forces, if you start from $\theta /2$ and go to 0, then your $d \theta$ is negative because it's decreasing so you have to add a minus sign $\endgroup$ – Brain Stroke Patient Jul 30 '20 at 18:07
  • $\begingroup$ Thank you Triatticus and Brain Stroke Patient for explaining.. i really appreciate your help $\endgroup$ – Harshavardhan Hajeri Jul 30 '20 at 18:14

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