Q.An unsymmetrical double convex thin lens forms the image of a point object on its axis. Will the position of the image change if the lens is reversed?

The key idea is that the lens maker's formula depends only on the radii of curvature and the refractive index, not on which side faces the object.

Reasoning:

1. For a thin lens in air, the focal length is given by $\frac{1}{f} = (\mu - 1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$, where $R_1$ and $R_2$ are the signed radii of the two surfaces.
2. Reversing the lens swaps $R_1$ and $R_2$. The term $\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$ becomes $\left(\frac{1}{R_2} - \frac{1}{R_1}\right) = -\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$. …

Why the focal length is unchanged on reversal

For a thin lens in air the lens maker's formula is

Take the original orientation of the (unsymmetrical) double convex lens with first-surface radius $+a$ (centre of curvature on the outgoing side) and second-surface radius $-b$:

Now reverse the lens. The face of magnitude $b$ meets the light first (radius $+b$), and the face of magnitude $a$ becomes the second surface (radius $-a$):

$\frac{1}{f'} = (\mu - 1)\left(\frac{1}{b} - \frac{1}{-a}\right) = (\mu - 1)\left(\frac{1}{b} + \frac{1}{a}\right)$ …

Method: Testing Whether a Lens Property is Invariant Under Reversal

This method applies to any question asking whether reversing a lens (or a curved refracting element) changes its focal length or the image it forms.

Steps

Step 1: Write the lens maker's formula with signed radii for the original orientation

Assign a sign to each radius using the New Cartesian Sign Convention, with light travelling in a fixed direction (say, left to right):

$R_1$ is the radius of the surface light meets first, $R_2$ of the surface it meets second.

Step 2: Reverse the lens and relabel which surface is "first"

When the lens is turned around, the surface that used to be second (radius $R_2$) is now first, and vice versa. Crucially, each surface keeps its own physical shape and orientation relative to its own centre of curvature — only the order in which light meets them changes. Write the new formula with the surfaces swapped:

where $R_2'$ and $R_1'$ are the same two physical surfaces as before, now met in the new order, with signs re-assigned consistently to the same convention.

Step 3: Compare — recognise the algebraic symmetry …