Consider the following statements about continuous charge distributions:
(A) The linear charge density \(\lambda\) of a wire is the charge per unit length of a small line element of the wire.
(B) In defining the surface charge density \(\sigma\), the area element \(\Delta S\) is very small on the macroscopic scale but big enough to include a very large number of electrons.
(C) The surface charge density \(\sigma\) takes into account the quantisation of charge at the microscopic level.
(D) The units of the volume charge density \(\rho\) are \(\mathrm{C/m^{2}}\).
Which of the above statements are incorrect? Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (C) and (D) only
On a charged body, the charges are too many and too close to locate one by one. So we describe them by a charge density. It is taken over an element that is small to the eye but holds a very large number of charges.
(A) The linear charge density \( \lambda \) of a wire is the charge per unit length of a small line element of the wire. — True. \( \lambda=\Delta Q/\Delta l \), the charge \( \Delta Q \) in a small line element \( \Delta l \) divided by its length. Its unit is \( \mathrm{C/m} \).
(B) In defining the surface charge density \( \sigma \), the area element \( \Delta S \) is very small on the macroscopic scale but big enough to include a very large number of electrons. — True. The element must hold very many electrons, so that \( \Delta Q/\Delta S \) gives a smooth value at each point.
(C) The surface charge density \( \sigma \) takes into account the quantisation of charge at the microscopic level. — False. \( \sigma \) is a smoothed, macroscopic quantity. It ignores the quantisation of charge and the gaps between charges at the microscopic level.
(D) The units of the volume charge density \( \rho \) are \( \mathrm{C/m^{2}} \). — False. \( \rho=\Delta Q/\Delta V \) is charge per unit volume, so its unit is \( \mathrm{C/m^{3}} \). The unit \( \mathrm{C/m^{2}} \) belongs to \( \sigma \).
NCERT (2026-27):
Continuous Charge Distribution – “The linear charge density λ of a wire is defined by”
\( \displaystyle \lambda=\frac{\Delta Q}{\Delta l} \)\( \qquad (1.24) \)
Continuous Charge Distribution – “It is more feasible to consider an area element ΔS … on the surface of the conductor (which is very small on the macroscopic scale but big enough to include a very large number of electrons) and specify the charge ΔQ on that element.”
Continuous Charge Distribution – “The surface charge density σ so defined ignores the quantisation of charge and the discontinuity in charge distribution at the microscopic level.”
Continuous Charge Distribution – “… where ΔQ is the charge included in the macroscopically small volume element ΔV that includes a large number of microscopic charged constituents. The units for ρ are C/m3.”