Question

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

• A. K is negative, then liquid A and liquid B have convex meniscus.
• B. K is negative, then liquid A and liquid B have concave meniscus.
• C. K is negative, then liquid A has concave meniscus and liquid B has convex meniscus.
• D. K is zero, then liquid A has convex meniscus and liquid B has concave meniscus.

Hint:

The sign of $K$ indicates the nature of the meniscus of the liquids.

Answer 1

The Correct Option is C

Given two liquids A and B with contact angles \( \theta_A \) and \( \theta_B \) in a capillary tube, the ratio \( K \) is defined as \( K = \frac{\cos \theta_A}{\cos \theta_B} \). We aim to determine the meniscus shape (concave or convex) for liquids A and B when \( K \) is negative or zero.

Concept Used:

The meniscus shape is determined by the contact angle \( \theta \):

• For \( \theta < 90^\circ \), \( \cos \theta > 0 \), resulting in a concave meniscus (liquid wets the surface).
• For \( \theta > 90^\circ \), \( \cos \theta < 0 \), resulting in a convex meniscus (liquid does not wet the surface).

Step-by-Step Solution:

Step 1: Analyze the expression \( K = \frac{\cos \theta_A}{\cos \theta_B} \). The sign of \( K \) is contingent upon the signs of \( \cos \theta_A \) and \( \cos \theta_B \).

Step 2: When \( K \) is negative, \( \cos \theta_A \) and \( \cos \theta_B \) must have opposite signs. This implies:

• One liquid has \( \theta < 90^\circ \) (concave meniscus).
• The other liquid has \( \theta > 90^\circ \) (convex meniscus).

Step 3: To determine which liquid corresponds to which case when \( K \) is negative:

• If \( \cos \theta_A > 0 \), Liquid A has a concave meniscus.
• If \( \cos \theta_B < 0 \), Liquid B has a convex meniscus.

Step 4: When \( K = 0 \), the condition \( K = 0 \) implies \( \cos \theta_A = 0 \). Therefore, \( \theta_A = 90^\circ \). At this angle, the meniscus is flat, indicating that Liquid A neither wets nor repels the surface. The nature of Liquid B's meniscus depends solely on \( \theta_B \).

Final Computation & Result:

The conclusive interpretation is:

\[ \boxed{\text{If } K \text{ is negative, then liquid A has concave meniscus and liquid B has convex meniscus.}} \]

Final Answer: The correct statement is — If \( K \) is negative, then liquid A has concave meniscus and liquid B has convex meniscus.