Electric Field At Centre Of Non Uniform Ring 
In the above solved example why only the cos components have been taken for calculation of electric field at centre ? Why not the sine components ? BTW in case you say that the sine components cancel out, please tell me why will they cancel out?
SOURCE:http://www.youtube.com/watch?v=kubLfWCIWv0 (I came across this video just sometime ago and had the above doubt.So please help!!)
 A: The sin components actually will cancel out because in the integral you will have $\cos\theta\sin\theta$ which is equal to $\sin(2\theta )/2$,which if you integrate from $0$ to $2\pi$ will give you zero.
I do not know how to explain it intuitively,i think that you can not predict this result by "physics intuition",but only with mathematics intuition(some might predict it based on the cos function of the charge distribution).
So, don't fret about it.
A: There is an intuitive reason why the electric field cancels at the center. Notice that the charge distribution is a cosine function, which means that if you stay at the center and watch at any specific direction, the charge at the opposite direction will have the opposite sign, and will cancel out the net force. This happens for any arbitrary direction of your choice.
A: ![enter image description here](https://i.stack.imgur.com/O1Sat.jpg
A: Looks like the answers are missing physical intuition, so I will try to explain it intuitively.
Also, note that this works only because the linear charge density is fucntion of cosine, had it been a complex one, this approach would become more hectic that using integration.
So, we know cosine function is positive in 1st and 4th quadrants and negative otherwise. Hence the ring will have positive and negative charges in the respective quadrants. Let the test charge at the centre be positive. Notice how the left side will try to pull the charge towards left while the right side will push it, again to the left. So the x component of electric field doesn't cancel. Now you can clearly see why the y component cancels out.
I know I am late but hopefully this will help someone.
