Image Formation at a Spherical Refracting Surface

Question

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 $R$?

Answer 1

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.

The equation changes slightly when you use a different sign convention (as shown here) but the conclusions will be the same.