A charge of \(16\ \mu\mathrm{C}\) is spread uniformly through the volume of a non-conducting cube of side 10 cm. The charge contained in a small cube of side 5 cm inside it is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(2\ \mu\mathrm{C}\)
The charge is spread uniformly, so the volume charge density \( \rho \) is the same at every point of the cube. The charge in any part of the cube is \( \rho \) times the volume of that part.
Side of the large cube, \( a=10\ \mathrm{cm}=0.1\ \mathrm{m} \)
Side of the small cube, \( b=5\ \mathrm{cm}=0.05\ \mathrm{m} \)
\( \begin{aligned} \rho &= \frac{Q}{a^{3}} \\ &= \frac{16\times10^{-6}}{(0.1)^{3}} \\ &= 1.6\times10^{-2}\ \mathrm{C\,m^{-3}} \end{aligned} \)
\( \begin{aligned} q &= \rho\,b^{3} \\ &= 1.6\times10^{-2}\times(0.05)^{3} \\ &= 1.6\times10^{-2}\times1.25\times10^{-4} \\ &= 2\times10^{-6}\ \mathrm{C}=2\ \mu\mathrm{C} \end{aligned} \)
The small cube has one-eighth of the volume of the large cube, so it holds one-eighth of the charge.
NCERT (2026-27):
Continuous Charge Distribution – “… The volume charge density (sometimes simply called charge density) is defined in a similar manner:”
\( \displaystyle \rho=\frac{\Delta Q}{\Delta V} \)\( \qquad (1.25) \)
Continuous Charge Distribution – “… Divide the charge distribution into small volume elements of size ΔV. The charge in a volume element ΔV is ρΔV.”