Problem 36 - Master Resource Book in JEE Main Physics

Question

A frame made of a metallic wire enclosing a surface area \(A\) is covered with a soap film. If the area of the frame of metallic wire is reduced by \(50 \%\), the energy of the soap film will be changed by (a) \(100 \%\) (b) \(75 \%\) (c) \(50 \%\) (d) \(25 \%\)

Step-by-Step Solution

Answer

(c) 50%

Step 1: Understand the Problem

We need to determine how the energy of a soap film changes as the area of its frame, made of a metallic wire, is reduced by 50%. Soap films have surface energy that is proportional to the surface area, given by the formula \( E = ext{Surface Tension} \times A \), where \( A \) is the area.

Step 2: Relation Between Energy and Area

The energy \( E \) of the soap film is proportional to its area. If the initial area is \( A \), and the energy is \( E \), then reducing the area by 50% means the new area \( A' = 0.5A \). Thus, the new energy \( E' = ext{Surface Tension} \times 0.5A = 0.5E \).

Step 3: Calculate the Percentage Change in Energy

The percentage change in energy can be calculated using the formula: \[ \text{Percentage Change} = \frac{E' - E}{E} \times 100\% \]. Substituting \( E' = 0.5E \), we get \( \text{Percentage Change} = \frac{0.5E - E}{E} \times 100\% = -50\%\).

Step 4: Determine the Correct Option

A change of \(-50\%\) means the energy is reduced by 50%. Therefore, option (c) \(50\%\) is the correct answer, as it explains the reduction in energy when the area is reduced by 50%.

Key Concepts

Surface Tension

Surface tension is an essential concept when discussing the properties of surfaces, particularly in liquids. It is the force that acts on the surface of a liquid, minimizing its surface area. Imagine a stretched elastic sheet, which pulls back towards a smaller area; this tugging is similar in nature to surface tension. It affects how liquids behave on surfaces, including how droplets form and spread.

In the case of soap films, surface tension plays a pivotal role in determining the film's stability and shape. Soap films are known to form minimal surfaces, meaning they try to occupy the smallest possible area to limit the overall energy. This is due to the continuous action of surface tension seeking to reduce surface area. When considering the energy of a soap film, one must remember that this energy is directly proportional to its surface area since it equals the surface tension coefficient times the area of the film.

Surface Area Reduction

When dealing with systems like a soap film bound by a metallic wire frame, reducing the surface area significantly impacts the system's properties. Considerable reductions, such as a 50% decrease in area, offer a striking example. With soap films, reducing their surface area inevitably causes a reduction in their stored energy.

This directly relates to the energy formula for a soap film, where energy is derived from multiplying the surface tension by the area. If the surface area is reduced, as in our exercise where the area is decreased by half, you effectively cut the energy to half its original value.

• An initial area of 1 unit with a corresponding energy of 1 unit would, after a 50% reduction, result in a new area and energy of 0.5 units each.
• Such a significant reduction highlights the direct proportionality between area and energy within soap films.

It is fascinating to note how tangible an impact area changes can have on the physical properties of a system.

Soap Film Energy Change

The energy contained within a soap film is not static; it fluctuates with changes in the film's size. Originally, the energy is calculated using the expression: energy equals surface tension times area. Hence, any alteration in area will directly affect the energy.

In the scenario of our exercise, the energy change follows a clear path. Initially, with a full area (say, 1 unit), the film has full energy (also 1 unit). When the area is cut by 50%, the resulting energy is also half, equating to 0.5 of the initial energy. To understand the percentage change, one calculates the difference in energy before and after the reduction.

• From original to reduced, the equation is: Percentage Change = ((new energy - original energy)/original energy) x 100%.
• Substituting values gives: Percentage Change = ((0.5E - E)/E) x 100% = -50%.

Here, the negative percentage change signifies a reduction, underlining the direct impact area reduction has on energy. Understanding this concept ensures clarity in numerous practical applications, like those found in industrial and laboratory settings involving films and membranes.