A whin wire ring of radius `r` carries charge `q`. Find the magnitude of the electric field strength on the axis of the ring as a function of distance `l` from its centre. Investigate the obtained function at `l gt gt r`. Find the maximum strength magnitude and the corresponding distance `l`. Draw the appoximate polt of the function. `E (l)`.

Explanation

See figure (Modify for maximum E) We know due to ring electric field strength at a distance l from its centre on its axis can be given as

`E=(Kql)/((l^2+r^2)^(3//2))" "...(1)`
For `lgtgtr,` we have `E=1/(4piepsilon_0)xxq/(l^2)`
Thus the ring behaves like a point charge.
For `E_("max")(dE)/(dl)=O`. From equations we get
`(dE)/(dl)=q/(4piin_0)[((r^2+l^2)^(3//2).1-(3l)/(2)(r^2+l^2)^(1//2)xx2l)/((r^2+l^2)^3)]=0`
or `(r^2+l^2)^(3//2)=3/2(r^2+l^2)^(1//2)xx2l^2`
Solving we get, `l=r/(sqrt(2))" "...(2)`
Substituting the value of  in equation (1) we get
`E=(kq(r+sqrt(2)))/((r^2+r^2//2)^(3//2))=(2kq)/(3sqrt(3)r^2)`