Why does the current in a purely capacitive AC circuit lead the voltage by 90 deg? 1.When we mathematically derive the expression for the current from a sinusoidal voltage source (v=V sin(wt)), we take the derivative of q=cv where c is the capacitance. The final expression we get is i=I cos(wt) which we express as i=I sin (wt+90). We can very well express it as i=I sin (wt-90). So we can say that the current lags the voltage instead of saying what the standard is.


*What makes the current lead the voltage? If I have a manual voltage source which I can use to change the voltage across the capacitor as and when I desire (assuming no resistance), will the current still lead? How and why?

 A: Both your questions relate to the special property of the sine wave, whose derivative is another sine wave shifted ahead by a quarter period, or as you write, 90 degrees, in advance.
You are right we can express the current as "i=I sin (wt+90)". But this is a different function than "i=I sin (wt-90)", which would have a wrong sign. Therefore it is only correct to say the current leads the voltage. 
Everything above holds only for sine waves. If you have an arbitrary voltage source, the current has to be computed using derivatives of the waveform.
A: Your question is very reasonable. Indeed both solutions are allowed from a mathematical point of view. But at any time you check current and voltage with an oscilloscope you will get the same shift.
This has to do with the asymmetry of the Lorentz force for negative and positive charged particles. In a current carrying coil the magnetic field will be oriented always in the same direction for a current of electrons. But if it would be possible for you to have a current of positrons (in a world of antimatter) or of protons (ionised liquid in a spiral tube) you will get a magnetic field which will be in the opposite direction to the usual direction.
What that has to do with the shift of current and voltage in an alternating current? The point is that an alternating current is made by induction from a varying magnetic field (the opposite process of the Lorentz force). The magnet moves the electrons. If a magnet with the same direction of his magnetic field and the same rotational direction would induce a current of positrons in a coil (same direction of winding), you would see on your oscilloscope a shift of the waves you described as alternative to the usual result. 
A: Since the voltage on the capacitor comes from a buildup of current, do you find it surprising that the volatage lags the current? But that's the same as saying the current leads the voltage.
