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.