Continuous Charge Distribution

In a continuous charge distribution, charge is treated as continuously spread along a line, over a surface, or throughout a volume. Such distributions are described by linear, surface, and volume charge densities, respectively. A charge concentrated at a single point is a point charge, so option A does not represent a continuous distribution.

Linear charge density represents the charge distributed per unit length on a wire or rod. It is given by \(\lambda = \frac{dq}{dl}\), so the correct answer is length. Charge per unit area is surface charge density, whereas charge per unit volume is volume charge density.

Surface charge density is represented by σ and is defined as charge distributed per unit area: σ = Q/A for a uniform distribution. Its SI unit is coulomb per square metre (C m⁻²). Hence it is associated with area, not length, volume or speed. Option B is correct. Charge per unit length is linear density, while charge per unit volume is volume density.

Volume charge density is used when charge is distributed through a three-dimensional region. It is expressed as
ho = rac{dQ}{dV}
, meaning charge per unit volume. Its SI unit is
ext{C/m}^3
. Charge per unit area is surface charge density, not volume charge density.

The radius of a thin wire is negligible compared with its length, so the wire is modeled as a line. Therefore, its charge is described by linear charge density: \(\lambda = \frac{dq}{dl}\). Its SI unit is \(\mathrm{C\,m^{-1}}\). Surface charge density is used for surfaces, whereas volume charge density is used for three-dimensional bodies.

Charge on a sheet is distributed over a two-dimensional surface, so it is described by surface charge density: \(\sigma = \frac{dQ}{dA}\), where \(dA\) is a small surface area. Linear charge density, \(\lambda = \frac{dQ}{dl}\), is used for one-dimensional distributions such as a wire, whereas volume charge density, \(\rho = \frac{dQ}{dV}\), is used for charge spread through a volume.

When charge is distributed throughout the interior volume of a solid sphere, it is described by volume charge density:

dq = \rho\,dV, where \rho is charge per unit volume. Linear charge density applies to one-dimensional distributions such as a wire, while surface charge density applies when charge lies only on a surface. Exam tip: Words such as “throughout the interior” or “spread in volume” indicate volume charge density, \rho.

Linear charge density is the charge per unit length on a charged wire or line. It is expressed as \(\lambda=\frac{dq}{dl}\), and its SI unit is \(\mathrm{C\,m^{-1}}\). Charge per unit area is surface charge density, whereas charge per unit volume is volume charge density.

Surface charge density means the amount of electric charge present on each unit of surface area. For a uniform distribution, it is written as σ = Q/A, where Q is charge and A is area; its SI unit is C/m². Therefore option B is correct. Option A describes linear charge density, option C describes volume charge density, and option D is not a standard charge-density definition.

Volume charge density is the charge present per unit volume of a charge distribution. It is expressed as \\(\rho = \frac{dq}{dV}\\), and its SI unit is \\(\mathrm{C\,m^{-3}}\\). Charge per unit area is surface charge density, denoted by \\(\sigma\\), so option B is not correct.

In a uniform charge distribution, equal lengths, equal areas, or equal volumes contain equal amounts of charge. Therefore, the relevant charge density is constant:
ext{λ}=rac{dq}{dl} for a line distribution,
ext{σ}=rac{dq}{dA} for a surface distribution, and
ext{ρ}=rac{dq}{dV} for a volume distribution. Option C represents a non-uniform distribution because the charge density varies with position.

In a non-uniform distribution, charge is not spread equally throughout the relevant length, area or volume. Consequently, the density is a function of position, such as λ(x), σ(x) or ρ(x), and may be larger in one region and smaller in another. Thus option B is correct. A describes a uniform distribution, whereas C and D are unsupported absolute claims.

In a continuous charge distribution, charge is spread along a line, over a surface, or through a volume. It is divided into small elements \(dq\) so that the electric-field contribution of each element can be found and integrated to obtain the total electric field. Unlike option B, the purpose of using \(dq\) is not to make the total charge zero.

A continuous charge distribution is divided into infinitesimal charge elements \(dq\). The electric field \(d\vec{E}\) due to each element is found, and all these fields are added vectorially through integration. This is the principle of superposition. Conservation of charge concerns the total charge, not the addition of field contributions. Exam tip: For a continuous distribution, writing \(d\vec{E}\) and integrating is based on superposition.

In a continuous charge distribution, charge is treated as infinitesimal elements \(dq\). Integrating these elements over the entire distribution gives the total charge: \(Q=\int dq\). For example, for a linear distribution, \(dq=\lambda\,dl\). Taking the charge density at only one point does not give the charge of the whole distribution. Exam tip: When you see “continuous distribution” and “total charge”, recall \(Q=\int dq\).

A uniformly charged ring is modeled as a continuous line distribution in which equal lengths of the circumference carry equal charge. If the total charge is Q and the ring radius is R, its constant linear charge density is λ = Q/(2πR). Thus the charge lies along the complete circumference, not at the centre or at one isolated point. Option D would describe a non-uniform ring.

The average linear charge density is
\(\lambda_{\text{avg}}=\frac{Q}{L}\), where \(Q\) is the total charge and \(L\) is the length of the wire. If \(Q\) remains constant while \(L\) increases, \(\lambda_{\text{avg}}\) decreases. Therefore, option B is correct. It is not zero unless the total charge \(Q\) is zero.

Average surface charge density is defined as σ = Q/A, where Q is the total charge and A is the surface area. When Q remains constant while A increases, the same charge is spread over a larger area, so the quotient Q/A becomes smaller. Therefore the average surface charge density decreases. It does not necessarily become half unless the area specifically doubles; options A and D contradict the inverse relation.

The magnitude of the average volume charge density is \(\lvert\rho_{\text{avg}}\rvert=\dfrac{\lvert Q\rvert}{V}\). When \(\lvert Q\rvert\) is constant and \(V\) increases, the same charge is distributed through a larger volume, so \(\lvert\rho_{\text{avg}}\rvert\) decreases. It becomes zero only if the total charge is zero or the volume is taken to be infinite.

A continuous charge distribution is divided into infinitesimal parts for calculation. The charge present on one such infinitesimal part is denoted by \(dq\). Integrating all charge elements gives the total charge: \(Q=\int dq\). Option C represents charge per unit length, namely the linear charge density \(\lambda\), rather than a charge element itself.

When charge is spread along a wire, over a surface, or through a volume and the dimensions of the charged object are significant, the charge is divided into small elements \(dq\) and their effects are added. This is a continuous charge distribution, so option A is correct. In contrast, if charge is concentrated in a very small region or the observation point is very far compared with the object's size, the object can often be approximated as a point charge.

When the observation distance is much greater than the object's size, the separate positions of charges within the object become negligible. Thus, its total charge may be regarded as concentrated at one point, so the object is treated as a point charge. Uniform volume or surface charge distributions describe the actual spread of charge in an extended body, whereas an electric dipole requires the separation of equal and opposite charges to be significant.

Charge density describes how charge is distributed in a region. For linear, surface, and volume distributions, it gives charge per unit length, area, and volume respectively. For example, volume charge density is
\(\rho = \frac{dQ}{dV}\). Therefore, option A is correct. Total charge states only the total amount of charge, whereas charge density gives information about its spatial distribution.

Linear charge density is

dfrac{Q}{l}

, meaning charge per unit length. Hence, its SI unit is coulomb per metre (C/m). Coulomb per square metre is the unit of surface charge density, while coulomb per cubic metre is for volume charge density. Exam tip: divide by m, m² and m³ for linear, surface and volume charge densities respectively.

Surface charge density represents charge distributed per unit area, so it is defined by σ = Q/A. Charge Q is measured in coulombs and area A in square metres; therefore the SI unit is coulomb per square metre, written C/m². Coulomb per metre is the unit of linear charge density, while C/m³ belongs to volume charge density. Metre per coulomb is the reciprocal type of unit and is not appropriate here.