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JEE Advanced2019PhysicsElectrostaticsActual

A thin spherical insulating shell of radius R carries a uniformly distributed charge such that the potential at its surface is V 0 . A hole with a small area α 4 π R 2 α ≪ 1 is made on the shell without affecting the rest of the shell. Which one of the following statements is correct?

Options

  1. AThe magnitude of electric field at the center of the shell is reduced by α V 0 2 R
  2. BThe potential at the center of the shell is reduced by 2 α V 0
  3. CThe ratio of the potential at the center of the shell to that of the point at 1 2 R from center towards the ho
  4. DThe magnitude of electric field at a point, located on a line passing through the hole and shell's center on a

Correct answer

C. The ratio of the potential at the center of the shell to that of the point at 1 2 R from center towards the ho

Step-by-step solution

Let ‘ A ’ be the point at a distance 2 R from center and ‘ B ’ be the point at a distance of R 2 from center. Now, let total charge on spherical shell distributed uniformly = Q ⇒ C h a r g e p e r u n i t a r e a = Q 4 π r 2 Charge on elemental area ‘ d A ’ . Given potential at surface = V 0 ⇒ V 0 = K Q R Now, potential at center of the shell = V C V C = K Q R - K ( d Q ) R = K Q R - K ( α Q ) R = K Q R ( 1 - α ) ⇒ V C = V 0 1 - α ⇒ Reduced b y α V 0 Now potential = V B ⇒ V B = K Q R - K a Q R / 2 = V 0 1 - 2 a ⇒ R

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