Let a total charge 2Q be distributed in a sphere of radius R, with the charge density given by ρ(r)

Let a total charge 2Q be distributed in a sphere of radius R ……

Question 2:

Let a total charge 2Q be distributed in a sphere of radius R, with the charge density given by ρ(r) = kr, where r is the distance from the centre. Two charges A and B, of −Q each, are placed on diametrically opposite points, at equal distance, a form the centre. If A and B do not experience any force, then:

Solution:

Let a total charge 2Q be distributed in a sphere of radius R, with the charge density

Two charges kept at point A and B will not experience any force under equilibrium condition, net electrostatic force on system must be equal to zero.

$vec{F_AC} + vec{F_AB} = 0$
${F_AC} = {F_AB}$ ………….. (1)
Where,
$vec{F_AC} = Force on charge A due to charged sphere$
$vec{F_AB} = Force between charge A and B$

F_AC = QE
where E = Electric field due to charged sphere at point A

Find E= ? Using Gauss theorem,

$oint{}{} {E . dS} = Q_inside /epsilon_0$

$E. oint{0}{a} { dS} = Q_inside /epsilon_0$

$E. {4 pi (a-0)^2} = Q_inside /epsilon_0$

$E. {4 pi a^2} = Q_inside /epsilon_0$ …………. (2)

Find Qinside = ? the Gaussian surface having radius ‘a’

$Q_inside = int{}{} {rho . dV}$

Given rho = kr

putting this value in above equation,

$Q_inside = int{}{} {rho . dV} = int{}{} {kr.dV}$

Find dV= ? Volume of sphere is given by,

V = 4/3 pi r^3

differentiating this equation to find dV

$dV/dr = d/dr ({4/3 pi r^3})$

$dV/dr = {4/3 pi * 3r^{3-1}}$

$dV = {4pi r^2} dr$

So, $Q_inside = int{0}{V} {rho . dV} = int{0}{V} {kr.dV} = int{0}{a} {kr.{4pi r^2} .dr}$

$Q_inside = int{0}{a} {kr.{4pi r^2} .dr}$ ……………. (3)

$Q_inside = int{0}{a} {k.{4pi r^3} } .dr = {k.{4pi (a-0)^4 / 4} }$

$Q_inside = {k. {pi a^4} }$ …………………. (4)

Finding value of constant k = ? Since, total charge on sphere (Q surface) is 2Q. Using equation (3)

$Q_surface = int{0}{R} {kr.{4pi r^2} .dr} = {k.{4pi (R-0)^4 / 4} }$

$Q_surface = 2Q = {k. {pi R^4} }$

$k = {2Q}/{pi R^4}$

Putting value of k in equation (4)

$Q_inside = {k. {pi a^4} } = {2Q}/{pi R^4} . {pi a^4} = {2Q a^4}/{R^4}$

Putting value of Qinside to equation (2)

$E. {4 pi a^2} = Q_inside /epsilon_0 = ({{2Q a^4}/{R^4}})/epsilon_0$

$E = 1/{ 4 pi epsilon_0} {{2Q a^2}/{R^4}}$

So, $F_AC = QE = Q . {1/{ 4 pi epsilon_0} {{2Q a^2}/{R^4}}} = 1/{ 4 pi epsilon_0} {{2Q^2 a^2}/{R^4}}$

Find electrostatic force between charge A and charge B using Coulomb’s law, ${F_AB} = ?$

${F_AB} =1/{ 4 pi epsilon_0} {Q^2 / (2a)^2} =1/{ 4 pi epsilon_0} {Q^2 / {4a}^2}$

Now from equation (1), ${F_AC} = {F_AB}$

$1/{ 4 pi epsilon_0} {{2Q^2 a^2}/{R^4}} = 1/{ 4 pi epsilon_0} {Q^2 / {4a}^2}$

a^4 = 1/8 R^4

$a = {(1/8)^{1/4}} R$

So, option C is correct

Some questions

What is the SI unit of charge density?

The SI unit of charge density depends on whether it is volumetric or surface charge density. For volumetric charge density, the unit is coulombs per cubic meter (C/m³), while for surface charge density, the unit is coulombs per square meter (C/m²).

How does charge density affect electric field strength?

Charge density is directly proportional to the electric field strength. In other words, the greater the charge density, the stronger the electric field in the region around the charge.

How is charge density related to the Coulomb force?

The Coulomb force is the force of attraction or repulsion between two charged particles. The magnitude of the Coulomb force is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. Charge density is related to the Coulomb force because it affects the strength of the electric field, which in turn affects the force experienced by a charged particle.

What is the difference between volumetric charge density and surface charge density?

Volumetric charge density is the amount of charge per unit volume of a material, while surface charge density is the amount of charge per unit area of a surface. In other words, volumetric charge density refers to the charge contained within a given volume, while surface charge density refers to the charge contained on a given surface area.

Let a total charge 2Q be distributed in a sphere of radius R , Let a total charge 2Q be distributed in a sphere of radius R

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Let a total charge 2Q be distributed in a sphere of radius R ……

Some questions

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