JEE Advanced · 2009 · 12 Apr 2009 · Paper 1 · Electrostatics

Three concentric metallic spherical shells of radii

R, 2 R, 3 R

, are given charges

Q_{1}, Q_{2}, Q_{3}

, respectively. It is found that the surface charge densities on the outer surfaces of the shells are equal. Then, the ratio of the charges given to the shells,

Q_{1} : Q_{2} : Q_{3}

, is

(A)1 : 2 : 3
(B)1 : 3 : 5✓ Correct
(C)1 : 4 : 9
(D)1 : 8 : 18

Answer: (B)

Solution

The surface charge density on the outer surface of each shell is equal. For the innermost shell of radius

R

, the charge is

Q_{1}

distributed on its outer surface of area

4 π R^{2}

, so

σ = \frac{Q _{1}}{4 π R ^{2}}

. For the middle shell of radius

2 R

, the total charge on its outer surface is

Q_{1} + Q_{2}

distributed on area

4 π (2 R)^{2}

, so

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

. For the outermost shell of radius

3 R

, the total charge on its outer surface is

Q_{1} + Q_{2} + Q_{3}

distributed on area

4 π (3 R)^{2}

, so

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

. Equating these:

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

. From the first two:

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

, so

Q_{2} = 3 Q_{1}

. From the first and third:

9 Q_{1} = Q_{1} + Q_{2} + Q_{3}

, so

Q_{3} = 8 Q_{1} - Q_{2} = 8 Q_{1} - 3 Q_{1} = 5 Q_{1}

. Therefore,

Q_{1} : Q_{2} : Q_{3} = 1 : 3 : 5

JEE Advanced · 2009 · 12 Apr 2009 · Paper 1 · Single correct option

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JEE Advanced · 2009 · 12 Apr 2009 · Paper 1 · Electrostatics

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