Continuous Charge Distribution

For a uniform volume charge distribution, the volume density ρ is constant and the charge in any region is q = ρV. If the chosen region has volume V/3, then q = ρ(V/3) = (ρV)/3 = Q/3, with Q denoting the total charge. Therefore one third of the volume contains one third of the charge. This proportionality would not hold for a nonuniform density.

A circular ring is defined as the set of points lying at a fixed radius from its centre. Therefore, every small charge element located on the circumference has the same distance R from the centre. This equal-radius property is important when calculating the potential or field of a ring because corresponding elements have comparable geometrical distances. The distance is not zero or infinite, and it is not different for different points on the same ideal ring.

Charge density measures how much charge is contained per unit geometrical measure. Depending on the distribution, linear density is λ = dq/dl, surface density is σ = dq/dA, and volume density is ρ = dq/dV. A larger magnitude therefore means more charge in the same length, area, or volume, assuming the comparison uses the same type of density. It does not imply zero total charge, absence of charge, or greater mass.

Positive charge density means that the small source element has positive charge. A positive test charge placed near it is repelled, so the electric-field vector produced by that element points away from the element along the line joining source and observation point. The complete field of a distribution may have components that combine or cancel, but the elementary direction follows the source-charge sign. It is not always zero and is not necessarily tangential to a circumference.

For a uniform linear charge distribution, the linear charge density λ is constant and Q = λL. If a segment has length L/4, its charge is q = λ(L/4) = Q/4, where Q = λL is the charge on the complete length. Therefore one-fourth of the length contains one-fourth of the total charge. The other fractions would apply only if the selected length had the corresponding ratio, assuming the density remains uniform.

For a uniform surface distribution, the surface charge density σ is constant, so charge is proportional to area: Q = σA. If the selected area is 3A/4, its charge is q = σ(3A/4) = 3Q/4, where Q is the charge on the full surface. Thus three-fourths of the area contains three-fourths of the total charge. The result would differ only if the surface density varied from one region to another.

For a uniform volume distribution, the volume charge density remains constant, so charge is directly proportional to volume: Q = ρV. If the selected volume is V/4, its charge is q = ρ(V/4) = Q/4. Therefore, one-fourth of the total charge is obtained. The whole, half, and double-charge choices would require different volume ratios or a non-uniform density.

A linear distribution extends along one dimension, so its charge is described using linear charge density λ, whose unit is coulomb per metre. For a small length element dl, the charge is dq = λ dl. Therefore the relevant measure is a small length. A small area belongs to a surface distribution, a small volume to a volume distribution, and time is not the geometrical measure used here.

A surface distribution spreads charge over a two-dimensional region. Its surface charge density is σ, measured in coulombs per square metre. For a small surface element dA, the charge is dq = σ dA, so the relevant geometrical quantity is a small area. Length is used for a linear distribution, volume for a volume distribution, and temperature does not determine the differential charge element in this context.

A volume distribution occupies a three-dimensional region and is described by volume charge density ρ, measured in coulombs per cubic metre. For a small volume element dV, the differential charge is dq = ρ dV. Hence option C is correct. A length is appropriate for linear density and an area for surface density; speed is a mechanical quantity and is not the geometrical measure used to calculate charge in this distribution.

The denominator in a density unit identifies the geometrical measure over which charge is distributed. Coulomb per metre, C/m, means charge per unit length and is therefore the unit of linear charge density λ. Hence the distribution is linear. Surface density has unit C/m², while volume density has unit C/m³. Time distribution is not one of the charge-distribution types used here.

Coulomb per square metre, written C/m², represents charge divided by area. This is surface charge density, commonly denoted by σ, so the distribution is a surface distribution. Linear density uses C/m and volume density uses C/m³. The unit of force is newton, not coulomb per square metre, so force distribution cannot be the correct interpretation.

The unit C/m³ means that charge is specified per unit volume. This quantity is volume charge density, denoted by ρ, and it describes a volume distribution throughout a three-dimensional region. By comparison, C/m identifies a linear distribution and C/m² identifies a surface distribution. Direction is not a geometrical measure in a charge-density unit, so it cannot be the answer.

The governing relation for a uniform surface charge distribution is Q = σA, where σ is surface charge density and A is area. Let the original charge be Q = σA. After the changes, Q′ = (σ/2)(2A) = σA = Q. Thus, the reduction in density is exactly compensated by the increase in area, so option A is correct. Options B and C consider only one change, while D has no physical basis.

For a uniform linear charge distribution, total charge is Q = λL, where λ is linear charge density and L is length. If λ′ = 3λ and L′ = L/3, then Q′ = λ′L′ = (3λ)(L/3) = λL = Q. Therefore the two changes cancel exactly and the total charge remains unchanged, making option A correct. Options B and C ignore one factor, while D incorrectly multiplies both changes without considering the reciprocal length change.

For a uniform linear charge distribution, the total charge is obtained by multiplying linear charge density by length: Q = λL. Substituting λ = 4 C/m and L = 3 m gives Q = (4 C/m)(3 m) = 12 C. The metre units cancel, leaving coulombs. Hence option B is correct. Seven is an incorrect sum, while four and three use only one of the given quantities instead of the required product.

For a uniform surface charge distribution, total charge equals surface charge density multiplied by area: Q = σA. Using σ = 5 C/m² and A = 2 m², Q = (5 C/m²)(2 m²) = 10 C. The square-metre units cancel correctly, so option A is the answer. Five ignores the area, 2.5 divides instead of multiplying, and seven has no valid calculation from the supplied values.

For a uniform volume charge distribution, total charge is the volume density multiplied by the occupied volume: Q = ρV. With ρ = 3 C/m³ and V = 5 m³, Q = (3 C/m³)(5 m³) = 15 C. The cubic-metre units cancel, leaving coulombs. Thus option B is correct. Eight is an addition, whereas three and five merely repeat one given quantity rather than applying the required product.

Linear charge density means charge per unit length, so the governing relation is λ = Q/L. Substituting the total charge Q = 18 C and length L = 6 m gives λ = 18/6 = 3 C/m. Therefore option A is correct. Multiplying would produce the wrong physical quantity, while 6 and 18 result from using the length or total charge without performing the required division.

Surface charge density is defined as charge per unit area, so σ = Q/A. For Q = 20 C and A = 5 m², σ = 20/5 = 4 C/m². Hence option B is correct. The units show why division is required: coulombs divided by square metres gives C/m². Option A is half the correct value, option C confuses area with density, and option D ignores the area.

Volume charge density is charge per unit volume, described by ρ = Q/V. Substituting Q = 24 C and V = 8 m³ gives ρ = 24/8 = 3 C/m³. Therefore option B is correct. The cubic-metre denominator is essential because the charge is distributed throughout a volume. Option A is too small, while C and D simply copy one of the given numbers without applying the definition.

For a uniform wire, total charge is Q = λL. With linear density λ unchanged, Q is directly proportional to the wire length L. If the length changes from L to 3L, the new charge is Q′ = λ(3L) = 3λL = 3Q. Therefore option C is correct. Option A reverses the proportionality, option B ignores the length change, and option D incorrectly squares the factor of three.

A surface distribution spreads charge over a two-dimensional surface. Therefore, the elemental charge is written as dq = σ dA, where σ is surface charge density and dA is a small area element. A length element belongs to a line distribution, while a volume element belongs to a volume distribution; speed is unrelated. Hence option B is correct.

A volume distribution occupies a three-dimensional region, so charge is described per unit volume. Its elemental relation is dq = ρ dV, where ρ is volume charge density and dV is a small volume element. A length element applies to linear density and an area element to surface density; temperature is not a spatial measure here. Therefore option C is correct.

The denominator of a density unit identifies the geometrical measure over which charge is distributed. Coulomb per metre, C m⁻¹, means charge per unit length, so it is the linear charge density λ. Surface density has unit C m⁻² and volume density has unit C m⁻³. Thus the correct answer is option A, not B or C.