Consider a solid cube made up of insulating material having a uniform volume charge density. Assuming the electrostatic potential to be zero at infinity, the ratio of the potential at a corner of the cube to that at the centre will be

Question

Consider a solid cube made up of insulating material having a uniform volume charge density. Assuming the electrostatic potential to be zero at infinity, the ratio of the potential at a corner of the cube to that at the centre will be (A) 1:1 (B) 1:2 (C) 1:4 (D) 1:8

Tardigrade Admin · Accepted Answer

<img class="img-fluid question-image" alt="Solution" src="https://cdn.tardigrade.in/q/nta/p-5ux2ibrzr3aw.png" /> ρ - Charge density of the cube Vlc o r n e r- Potential at the corner of a cube of side l Vlc e n t r e- Potential at the centre of a cube of side l Vl / 2c e n t r e- Potential at the center of a cube of side (l/2) Vl / 2c o r n e r- Potential at the corner of a cube of side (l/2) By dimensional analysis Vlc o r n e r propto (Q/l)=ρ l2 Vlc o r n e r=4 Vl / 2c o r n e r But by superposition Vlc e n t r e=8Vl / 2c o r n e r Because of the centre of the larger cube lies at a corner of the eight smaller cubes of which it is made Therefore, (Vlc o r n e r/Vlc e n t r e)=(4 Vl / 2c o r n e r/8 Vl / 2c o r n e r)=(1/2)

Q. Consider a solid cube made up of insulating material having a uniform volume charge density. Assuming the electrostatic potential to be zero at infinity, the ratio of the potential at a corner of the cube to that at the centre will be

$1 : 1$

$1 : 2$

$1 : 4$

$1 : 8$