Two liquids A and B have $$\theta_A$$ and $$\theta_B$$ as contact angles in a capillary tube. If $$K = \cos\theta_A / \cos\theta_B$$, then identify the correct statement:

The shape of the liquid surface in a capillary tube depends upon the magnitude of the contact angle $$\theta$$ with the glass wall.

• If $$\theta \lt 90^{\circ}$$, the liquid wets glass. The meniscus is concave and $$\cos\theta$$ is positive.
• If $$\theta = 90^{\circ}$$, the liquid does not rise or fall. The meniscus is flat and $$\cos\theta = 0$$.
• If $$\theta \gt 90^{\circ}$$, the liquid does not wet glass. The meniscus is convex and $$\cos\theta$$ is negative.

For the two liquids A and B we are given $$K = \dfrac{\cos\theta_A}{\cos\theta_B}$$.

Case 1: $$K \gt 0$$
Both $$\cos\theta_A$$ and $$\cos\theta_B$$ have the same sign, so both menisci are of the same type (either both concave or both convex).

Case 2: $$K = 0$$
This is possible only when $$\cos\theta_A = 0$$. Hence $$\theta_A = 90^{\circ}$$ (flat meniscus for liquid A) while liquid B can have any meniscus depending on $$\theta_B$$.

Case 3: $$K \lt 0$$
The ratio is negative, so $$\cos\theta_A$$ and $$\cos\theta_B$$ have opposite signs. Exactly one liquid has $$\theta \lt 90^{\circ}$$ (concave) and the other has $$\theta \gt 90^{\circ}$$ (convex).
Let $$\cos\theta_A$$ be positive and $$\cos\theta_B$$ be negative (the reverse choice still makes $$K$$ negative, but only one option in the list matches a concave A and convex B). Hence

 • liquid A: $$\theta_A \lt 90^{\circ}$$ ⇒ concave meniscus
 • liquid B: $$\theta_B \gt 90^{\circ}$$ ⇒ convex meniscus

This situation is described by Option C (statement 3): “K is negative, then liquid A has concave meniscus and liquid B has convex meniscus.”

Therefore, the correct statement is Option C (3).