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If same amount of charge is given to an insulating cube of side 'a' and an insulating sphere of diameter 'a', then which of the body will have greater volume charge density:
1. Cube
2. Sphere
3. Both will have equal volume charge density
4. Can't say

1 Answer

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Correct Answer - Option 1 : Cube

CONCEPT:

Linear charge density:

  • It is defined as the quantity of charge per unit length.
  • Its SI unit is C/m.
  • If ΔQ charge is contained in the line element Δl, the linear charge density λ will be,

\(⇒ \lambda=\frac{Δ Q}{Δ l}\)

Surface charge density:

  • It is defined as the quantity of charge per unit area.
  • Its SI unit is C/m2.
  • If ΔQ charge is contained in the elemental area Δs, the surface charge density σ will be,

\(⇒ \sigma=\frac{Δ Q}{Δ s}\)

Volume charge density:

  • It is defined as the quantity of charge per unit volume.
  • Its SI unit is C/m3.
  • If ΔQ charge is contained in the elemental volume Δv, the volume charge density ρ will be,

\(⇒ \rho=\frac{Δ Q}{Δ v}\)

CALCULATION:

Given Side of the cube = a, Diameter of the sphere = a, ΔQC = ΔQS = Q

  • The volume of the cube is given as,

⇒ ΔvC = a3     ----(1)

  • The volume of the sphere is given as,

\(⇒ Δ v_S=\frac{4}{3}\pi R^3\)

Where R = radius of the sphere

\(⇒ Δ v_S=\frac{4}{3}\pi \left ( \frac{a}{2} \right )^3\)

\(⇒ Δ v_S=\frac{\pi a^3}{6}\)     ----(2)

By equation 1 and equation 2,

⇒ ΔvC > ΔvS

If ΔQ charge is contained in the elemental volume Δv, the volume charge density ρ will be,

\(⇒ \rho=\frac{Δ Q}{Δ v}\)

If ΔQ = constant

\(⇒ \rho\propto\frac{1}{Δ v}\)     ----(3)

  • By equation 3 it is clear that the volume charge density is inversely proportional to the volume when the charge is constant.
  • Since the volume of the cube is more than the sphere so the volume charge density of the cube will be more than the sphere. Hence, option 1 is correct.

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