Problem 43 - Master Resource Book in JEE Main Physics

Question

A water film is made between two straight parallel wires of length \(10 \mathrm{~cm}\) separated by \(5 \mathrm{~mm}\) from each other. If the distance between the wires is increased by \(2 \mathrm{~mm}\). How much work will be done? Surface tension for water is 72 dyne \(\mathrm{cm}^{-1}\). (a) \(288 \mathrm{erg}\) (b) \(72 \mathrm{erg}\) (c) \(144 \mathrm{crg}\) (d) \(216 \mathrm{crg}\)

Step-by-Step Solution

Answer

288 erg (option a).

Step 1: Understanding the Problem

We have a water film created between two parallel wires, each 10 cm long and initially separated by 5 mm. The wires are then moved to increase the separation by 2 mm. We need to calculate the work done, given the surface tension of water as 72 dyne/cm.

Step 2: Calculate Initial and Final Area of Water Film

Initially, the separation between the wires is 5 mm, so the area of each side of the film is \( A_{\text{initial}} = 10 \times 0.5 = 5 \text{ cm}^2 \). When the separation increases by 2 mm, the final separation becomes 7 mm, thus the final area is \( A_{\text{final}} = 10 \times 0.7 = 7 \text{ cm}^2 \). Since the film has two sides, we need to double these areas.

Step 3: Calculate the Change in Surface Area

The change in the surface area due to the movement of the wires is calculated as \( \Delta A = 2 \times (A_{\text{final}} - A_{\text{initial}}) = 2 \times (7 - 5) = 4 \text{ cm}^2 \).

Step 4: Calculate the Work Done

Work is done against the surface tension, which is given by the formula \( W = \gamma \times \Delta A \), where \( \gamma = 72 \text{ dyne/cm} \) is the surface tension.\[ W = 72 \times 4 = 288 \text{ dyne cm} = 288 \text{ erg} \]

Step 5: Select the Correct Answer

The work done to increase the distance between the wires is 288 erg, which corresponds to option (a) 288 erg.

Key Concepts

Work Done in Physics

In physics, work is defined as the energy transferred to or from an object via the application of force along a displacement. It is calculated by the formula:

• \( W = F \times d \), where \( F \) is force and \( d \) is displacement.

When discussing surface tension, work is done to increase the surface area of a liquid. Here, the focus is on how energy is needed to stretch a liquid film, like that between two wires.
The work done can be likened to the force the liquid's surface tension applies over a given distance. This results in an increase in the film's area. For the given exercise, the work done is calculated using the formula:

• \( W = \gamma \times \Delta A \)

where \( \gamma \) represents surface tension (72 dyne/cm for water here) and \( \Delta A \) is the change in surface area.

Surface Area Change

The concept of surface area change is critical in problems involving thin films. Initially, in the exercise, the water film has a certain surface area based on the separation between two wires. The increase in separation leads to an increase in the surface area of the water film.

• Initially, we calculate the surface area with the formula: \( A = \, \text{length} \times \text{width} \).
• With two sides considered, we double the calculated area.

An increase in distance between the wires changes the width (from 5 mm to 7 mm in the example). This new area must be recalculated to understand how much additional work against surface tension is required.
This change in area is essential to determine the work done to increase the film's surface, by utilizing the difference \( \Delta A \) in initial and final surface areas.

Parallel Wires in Physics

Parallel wires are often used within physics to demonstrate concepts related to forces, current flow, and tension within different contexts. In this particular situation, they serve to suspend a water film.

• When placed parallel, they create a frame—holding the water film in the gap between them.
• The distance between these wires is crucial as it directly affects the film's surface area.

This results in changes in the physical properties of the film as the wires move further apart. Understanding this setup aids in conceptualizing how the film behaves, making it simpler to calculate work done when the separation changes. When the wires are pushed further apart, the film stretches, increasing the work done as discussed in earlier sections.