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let the total charge on the ring is Q then
electric field at the axis of a charged ring is given by
E=kQx/(R2 + x2)3/2
to find maximum electric field we can use the concept of maxima and minima....
dE/dx = d/dx of {kQx/(R2 + x2)3/2 }
=kQ{ 1 .(R2 +x2)3/2 - 3/2 (R2 +x2)1/2. x }/(R2 +x2)3 (by applying quotient rule)
now on putting dE/dx =0 ,we get
x2 -3x/2+R2 =0
x2 -3x/2 +9/16 -9/16 +R2 =0
(x-3/4)2 = 9/16-R2
x={(9-16R2)1/2 +3 }/4 units
therefore at a distance x from center of the ring electric field is maximum
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