# Image Formation at a Spherical Refracting Surface

I was thinking about refraction at spherical refracting surfaces and what the required conditions be for the formation of a real or virtual image?

I thought I could use this formula: $$\begin{equation*} \frac{n_2}{v} - \frac{n_1}{u} = \frac{n_2-n_1}{R} \end{equation*}$$

but wasn't sure if this was the right approach and how to proceed for various cases. ($$n_1$$ and $$n_2$$ are refractive indices of the two mediums, $$u$$ is the object distance, $$v$$ is the image distance and $$R$$ is the Radius of the sphere)

As a specific case, I was interested to find out whether a real or virtual image will be formed if an object is kept at a distance $$R$$ from a convex refracting surface made of glass and having Radius of Curvature $$R$$?

The equation you mentioned works with Cartesian sign convention: treat the vertex (centre of the spherical surface separating $$n_1$$ and $$n_2$$) as the origin. All distances measured to the left are negative and to the right positive, and that includes object distance $$u$$, image distance $$v$$ and the radius of curvature $$R$$.
Since this is a refraction phenomenon, a real image is formed when the object and the image are in different media, i.e., $$u$$ and $$v$$ have opposite signs. When they have the same sign, the image is virtual.