A wire having a uniform linear charge density  $\lambda$ is bent in the from of a ring of radius R. Point A as shown in the figure, is In the plane of the ring but not at the center. Two elements of the ring of lengths $a_1\text{ and }{a}_2$subtend very small same angle at the point A They are at distances ${\mathbf{r}}_1\text{ and }{r}_2$ from the point A respectively 

 

Let charge on $a_1\mathrm{\&}{a}_2\text{ are }{q}_1\mathrm{\&}{q}_2$ respectively  $q_1=\lambda a_1\mathrm{\&}{q}_2=\lambda a_2$

Electric field at A due to $a_1$ 

$E_1=\frac{q_1}{4\pi {ϵ}_0{r}_1^2}=\frac{\lambda \left(r_1\theta \right)}{4\pi {ϵ}_0{r}_1^2}$

Electric field at A due to $a_2$

$E_2=\frac{q_2}{4\pi {ϵ}_0{r}_2^2}=\frac{\lambda \left(r_2\theta \right)}{4\pi {ϵ}_0{r}_2^2}$

Potential at A due to $a_1$

$V_1=\frac{q_1}{4\pi {\epsilon }_0{r}_1}=\frac{\lambda \left(r_1\theta \right)}{4\pi {ϵ}_0{r}_1}$

Potential at A due to $a_2$

$V_2=\frac{q_2}{4\pi {ϵ}_0{r}_2}=\frac{\lambda \left(r_2\theta \right)}{4\pi {ϵ}_0{r}_2}$

Direction of electric field at A depends upon the nature of charge i.e. positive or negative .