The work done in blowing a soap bubble of \(10 \mathrm{~cm}\) radius is [surface tension of soap solution is \(\\{(3 / 100) \mathrm{N} / \mathrm{m}\\}\) ] (A) \(75.36 \times 10^{-4}\) Joule (B) \(37.68 \times 10^{-4}\) Joule (C) \(150.72 \times 10^{-4}\) Joule (D) \(75.36\) Joule

The formula for the work done in inflating a bubble is: \[W = T \Delta A\] where \(W\) is the work done, \(T\) is the surface tension, and \(\Delta A\) is the change in surface area of the bubble. Step 2: Calculate the change in surface area

The surface area of a sphere is given by the formula \(A = 4\pi r^2\), where \(r\) is the radius. For a soap bubble, we need to consider both the inside and outside surfaces, so the total surface area of a soap bubble is \(A_\text{total} = 2(4\pi r^2) = 8\pi r^2\). The change in surface area after inflating the soap bubble to a radius of \(10 \mathrm{~cm}\) is: \(\Delta A = A_\text{final} - A_\text{initial}\) Since the initial surface area is zero (\(A_\text{initial}=0\)), the change in surface area is equal to the total surface area of the soap bubble: \(\Delta A = 8\pi r^2\) Substitute the given radius, \(r = 10 \mathrm{~cm}\), into the formula: \(\Delta A = 8\pi (10)^2 \mathrm{cm}^2\) \(\Delta A = 800\pi \mathrm{~cm^2}\) Step 3: Calculate the work done in blowing the soap bubble

Use the formula for the work done, and substitute the given surface tension, \(T = \frac{3}{100} \mathrm{N/m}\), and the calculated change in surface area into the formula: \(W = T \Delta A\) \[W = \left(\frac{3}{100}\mathrm{N/m}\right)(800\pi \mathrm{~cm^2})\] To convert from \(\mathrm{cm^2}\) to \(\mathrm{m^2}\), we need to multiply by \(\frac{1}{10,000}\): \[W = \left(\frac{3}{100}\mathrm{N/m}\right)(800\pi \cdot \frac{1}{10,000} \mathrm{~m^2})\] \[W = \frac{3}{100} \cdot \frac{800\pi}{10,000} \mathrm{N} \cdot \mathrm{m}\] Simplify the expression: \[W = \frac{12 \pi}{5} \mathrm{Nm}\] Step 4: Convert the work done into the desired units

Convert the result from \(\mathrm{Nm}\) (Joules) to \(\times 10^{-4}\) Joules: \[W = \frac{12 \pi}{5} \mathrm{Nm} \times \frac{1}{10^{-4}}\] \[W = 75.36 \times 10^{-4} \mathrm{J}\] The work done in blowing the soap bubble is 75.36 × 10⁻⁴ Joule, which corresponds to option (A).