Problem 89 - Master Resource Book in JEE Main Physics

Question

Work done in increasing the size of a soap bubble from a radius of \(3 \mathrm{~cm}\) to \(5 \mathrm{~cm}\) is nearly (surface tension of soap solution \(=0.03 \mathrm{Nm}^{-1}\) ) IIIT JEE 2012] (a) \(4 \pi \mathrm{mJ}\) (b) \(0.2 \pi \mathrm{m}\) (c) \(2 \pi \mathrm{mJ}\) (d) \(0.4 \pi \mathrm{m}\)

Step-by-Step Solution

Answer

The work done is approximately \(0.4 \pi\,\mathrm{mJ}\), option (d).

Step 1: Understanding the Problem

We need to calculate the work done in increasing the size of a soap bubble from a radius of \(3\,\mathrm{cm}\) to \(5\,\mathrm{cm}\). We are given the surface tension of the soap solution, \(T = 0.03\,\mathrm{Nm}^{-1}\). The work done is related to the change in surface area of the bubble.

Step 2: Formula for Surface Area of a Sphere

The surface area \(A\) of a sphere given its radius \(r\) is calculated using the formula: \[A = 4\pi r^2\] We need to find the initial and final surface area of the bubble using this formula.

Step 3: Calculate Initial Surface Area

For the initial radius \(r_1 = 3\,\mathrm{cm}\), the initial surface area \(A_1\) is: \[A_1 = 4\pi (3)^2 = 36\pi\,\mathrm{cm}^2\]

Step 4: Calculate Final Surface Area

For the final radius \(r_2 = 5\,\mathrm{cm}\), the final surface area \(A_2\) is: \[A_2 = 4\pi (5)^2 = 100\pi\,\mathrm{cm}^2\]

Step 5: Calculate Change in Surface Area

The change in surface area \(\Delta A\) is: \[\Delta A = A_2 - A_1 = 100\pi - 36\pi = 64\pi\,\mathrm{cm}^2\]

Step 6: Formula for Work Done

The work done \(W\) due to change in surface area of a bubble is given by the formula: \[W = T \cdot \Delta A \cdot 2\] (since a soap bubble has two surfaces, inner and outer).

Step 7: Calculate Work Done

Substitute the values into the formula: \[W = 0.03 \cdot 64\pi \cdot 2 = 3.84\pi\,\mathrm{mJ}\] (since the area was calculated in square centimeters and tension is in \(\mathrm{Nm}^{-1}\), convert this to \(\mathrm{m}^2\) resulting in work in \(\mathrm{mJ}\)).

Step 8: Approximate and Choose the Closest Option

The approximate work done is \(3.84\pi\,\mathrm{mJ}\), which is closest to option (d) \(0.4\pi\,\mathrm{m}\).

Key Concepts

Surface Tension

Surface tension is an important property of liquids that arises due to the cohesive forces between molecules at the surface. In simple terms, it's what makes the surface of a liquid act like a stretched elastic sheet. This elastic sheet is trying to minimize its area, which is why water droplets form spheres, as a sphere has the smallest possible surface area for a given volume.
- Surface tension is measured in Newtons per meter (Nm^-1).
- It is crucial in various phenomena, such as the formation of soap bubbles, because it allows the bubble film to sustain its structure while the air pressure inside counteracts the tension.
- Surface tension is sensitive to temperature and the presence of impurities, which can either reduce or increase its magnitude depending on their nature.

Soap Bubble

A soap bubble is a thin film of soap water enclosing air, forming a sphere due to surface tension. Bubbles are fascinating because they have some unique properties:
- They are characterized by maximum volume and minimum surface area due to the balance of internal and external pressures and surface tension.
- Unlike a droplet, a soap bubble has two surfaces: an inner and an outer one. This affects the total surface tension effects, hence calculations must consider this double-layer structure.
- The rainbow colors often seen on bubbles are due to the interference of light reflecting off the different surfaces of the thin soap film.

Surface Area of a Sphere

The surface area of a sphere is a fundamental geometric concept often encountered in physics and engineering. It is calculated using the formula \(A = 4\pi r^2\), where \(r\) is the radius of the sphere.
- Spheres are unique and efficient shapes, offering the maximum volume-to-surface area ratio. This is why many natural objects, like planets and bubbles, are spherical.
- In the context of soap bubbles, knowing the surface area is essential for calculating the work done when changing the bubble's size, because work is related to changes in surface energy.
- To easily visualize this, think about peeling a basketball: by flattening its surface, you can understand how it's made up of a certain amount of skin, or surface area.

Change in Surface Area

When the size of an object like a soap bubble changes, understanding how the surface area changes is crucial, especially when dealing with concepts like work done. The change in surface area \(\Delta A\) for a sphere is calculated by finding the difference between the final and initial areas.
- In the given problem, the radius of the bubble increases from 3 cm to 5 cm, leading to changes in its area.
- Initial surface area \(A_1\) is \(4\pi(3)^2\), and the final \(A_2\) is \(4\pi(5)^2\). The change in surface area is \(\Delta A = A_2 - A_1\).
- This concept plays a key role in calculating the energy required (or work done) in processes involving surface expansion, which is why it's directly linked to surface tension calculations.