consider a uniformly charged ring of radius R.Find the point on the axis where the elecrtic field is maximum.

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.......

Hello raju....

This is very simple...Just it needs a few steps of derivation...I am solving it with derivation...from the begining Let me know if u get stick with any line ....

It is :

Electric field due to:

 Charged ring of charge Q and radius R.

1) The Center=0.
Derivation:

Linear charge density (l) on ring =

Consider the field at center due to any element =

But the field due to point diametrically opposite = in opposite direction.
Net field at center = 0 (By symmetry)

2)      On the axis =

On axis of ring at distance x.
Derivation:
         
                                  Fig (5)

As obvious from the diagram the field component along the line gets added due to opposite element.

Now ...We have to find at which  x the field is maximum...so calculate dE/dx ...

dE/dx = Z [ (R^2 + x^2)^3/2  - 3/2(R^2+x^2)^1/2 . 2(x).(x) ] / [R^2+x^2]^3   ; where Z = Q/4pi*epsilon

For maximum let dE/dx = 0

then ..(R^2x +x^2)^3/2 - 3x^2(R^2+x^2)^1/2 = 0

(R^2 +x^2)^1/2 [ (R^2 +x^2) - 3x^2 ] = 0

hence ...R^2 = 2x^2

or .... x = + R/root(2)   or - R/root(2)

Means If ring is kept symmetrically in x-y plane with center at origin then..field will be maximum at ...a distance ..of +R/root(2) above the +z-axis...and below the negative z axis...

Is it clear

With regards

Yagya

askiitians_expert