Q.Figure 9.30 shows an equiconvex lens (of refractive index $1.50$) in contact with a liquid layer on top of a plane mirror. A small needle with its tip on the principal axis is moved along the axis until its inverted image is found at the position of the needle. The distance of the needle from the lens is measured to be $45.0\ \text{cm}$. The liquid is removed and the experiment is repeated. The new distance is measured to be $30.0\ \text{cm}$. What is the refractive index of the liquid?

When the inverted image coincides with the needle (autocollimation), the needle is at the focus of the mirror–lens system, so each measured distance equals a system focal length:

• Lens + liquid: $F = 45.0\ \text{cm}$
• Glass lens alone: $f_1 = 30.0\ \text{cm}$

Radius of the equiconvex lens ($n_g = 1.50$, radii $+R$ and $-R$):

Focal length of the liquid (plano-concave) lens, in contact with the glass lens:

$\frac{1}{f_2} = \frac{1}{F} - \frac{1}{f_1} = \frac{1}{45} - \frac{1}{30} = -\frac{1}{90} \;\Rightarrow\; f_2 = -90\ \text{cm}$ …

Principle — why the coincidence gives a focal length

With a plane mirror below the lens, rays that leave the needle, pass through the lens, reflect off the mirror, and return retrace their path only when they strike the mirror normally — that is, when they travel parallel between lens and mirror. This happens when the needle sits at the focus of the lens system, so the measured needle distance equals the focal length.

Step 1 — Focal lengths from the two measurements

• Glass lens + liquid layer together: $F = 45.0\ \text{cm}$
• Glass lens alone (liquid removed): $f_1 = 30.0\ \text{cm}$

Step 2 — Radius of the equiconvex glass lens

With $n_g = 1.50$ and surface radii $+R$ and $-R$, the lens maker's formula gives

so $R = f_1 = 30.0\ \text{cm}$.

Step 3 — Focal length of the liquid lens

The liquid fills the gap between the convex glass lens and the plane mirror, forming a plano-concave lens in contact with the glass lens. For thin lenses in contact, …

Method: Optical Autocollimation with a Liquid Lens on a Mirror

This method solves "needle experiment" problems where a lens (or a lens plus a liquid layer) rests on a plane mirror, and a needle's own inverted image coinciding with itself is used to extract a focal length.

Steps

Step 1: Recognise WHY coincidence gives a focal length

Light leaving the needle, passing through the lens system, reflecting off the plane mirror, and retracing its own path back to the needle can only happen if the rays strike the mirror travelling parallel to the axis — which happens only when the needle sits exactly at the focus of the whole lens(+liquid) system. So each measured needle-to-lens distance IS a system focal length, no further optics needed to establish that.

Step 2: Use the lens maker's formula on the part of the system you already know

Apply this to the glass lens alone (its shape and index are given) to pin down any unknown radius of curvature.

Step 3: Isolate the unknown component's focal length using the thin-lens combination rule

For two thin elements in contact, …