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Concepts Of Physics MCQ Edition [Volume 2]PhysicsGauss's Law

A charge Q is uniformly distributed throughout the material of a hollow sphere with inner and outer radii r₁ and r₂ , as shown in the figure. Determine the electric field at a point P positioned a distance x from the centre, given r₁

Options

  1. AQ 4 ₀ x^2
  2. BQ(x^3 - r₁^3) 4 ₀ x^2 (r₂^3 - r₁^3)
  3. CQ(x^2 - r₁^2) 4 ₀ x^2 (r₂^2 - r₁^2)
  4. DQ(x^3 + r₁^3) 4 ₀ x^2 (r₂^3 + r₁^3)

Correct answer

B. Q(x^3 - r₁^3) 4 ₀ x^2 (r₂^3 - r₁^3)

Step-by-step solution

The total charge Q is uniformly distributed in the volume between r₁ and r₂ . The volume of the material of the hollow sphere is: V = 4 3 (r₂^3 - r₁^3) The uniform volume charge density is: = Q V = Q 4 3 (r₂^3 - r₁^3) To find the electric field at a distance x ( r₁ The charge enclosed by this Gaussian surface is the charge within the volume from r₁ to x : q_ enc = 4 3 (x^3 - r₁^3) Substituting the value of : q_ enc = Q 4 3 (r₂^3 - r₁^3) 4 3 (x^3 - r₁^3) = Q ( x^3 - r₁^3 r₂^3 - r₁^3 ) According to Gauss's Law: E d A

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