Three concentric metallic spherical shells of radii R, 2R and 3R are given charges Q1, Q2, Q3 respectively. It is found that the surface charge densities on the outer surface of the shells are equal. Then the ratio of the charges given to the shells Q1: Q2: Q3 is

Question

Three concentric metallic spherical shells of radii R, 2R and 3R are given charges Q1, Q2, Q3 respectively. It is found that the surface charge densities on the outer surface of the shells are equal. Then the ratio of the charges given to the shells Q1: Q2: Q3 is (A) 1: 8: 18 (B) 1: 4: 9 (C) 1: 2: 3 (D) 1: 3: 5

Tardigrade Admin · Accepted Answer

Since surface charge densities on the outer surface of shells are same.

∴ σ = \frac{Q _{3} + Q _{2} + Q _{1}}{4 π ( 3 R ) ^{2}} = \frac{Q _{2} + Q _{1}}{4 π ( 2 R ) ^{2}} = \frac{Q _{1}}{4 π R ^{2}}

or

\frac{Q _{3} + Q _{2} + Q _{1}}{9} = \frac{Q _{2} + Q 1}{4}

....(i)

\frac{Q _{2} + Q _{1}}{4} = \frac{Q _{1}}{1}

..(ii)

Q_{1} + Q_{2} = 4 Q_{1}

or

Q_{2} = 3 Q_{1}

....(iii)
Also

\frac{Q _{3} + Q _{2} + Q _{1}}{9} = \frac{Q _{1}}{1}

Q_{3} + 4 Q_{1} = 9 Q_{1}

or

Q_{3} = 5 Q_{1}

...(iv)

∴ Q_{1} : Q_{2} : Q_{3} = 1 : 3 : 5

Q. Three concentric metallic spherical shells of radii $R, 2 R$ and $3 R$ are given charges $Q_{1}, Q_{2}, Q_{3}$ respectively. It is found that the surface charge densities on the outer surface of the shells are equal. Then the ratio of the charges given to the shells $Q_{1} : Q_{2} : Q_{3}$ is

1 : 8 : 18

1 : 4 : 9

1 : 2 : 3

1 : 3 : 5

Q. Three concentric metallic spherical shells of radii R,2R and 3R are given charges Q1​,Q2​,Q3​ respectively. It is found that the surface charge densities on the outer surface of the shells are equal. Then the ratio of the charges given to the shells Q1​:Q2​:Q3​ is

1 : 8 : 18

1 : 4 : 9

1 : 2 : 3

1 : 3 : 5

Q. Three concentric metallic spherical shells of radii $R, 2 R$ and $3 R$ are given charges $Q_{1}, Q_{2}, Q_{3}$ respectively. It is found that the surface charge densities on the outer surface of the shells are equal. Then the ratio of the charges given to the shells $Q_{1} : Q_{2} : Q_{3}$ is