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Two infinite parallel non-conducting sheets, each with a uniform surface charge density + , are placed at x = -a and x = a . A solid non-conducting sphere of radius R ( R < a ) with a uniform volume charge density + is centered at the origin. Which of the following best describes the variation of the x-component of the net electric field ( E_x ) along the x-axis?

Options

  1. AE_x is zero for |x| < R , decreases in magnitude as 1 x^2 for R < |x| < a , and exhibits a discontinuous jump
  2. BE_x increases linearly in magnitude for |x| < R , decreases in magnitude as 1 x^2 for R < |x| < a , and is con
  3. CE_x is exactly zero for |x| < a , and exhibits a discontinuous jump of magnitude ₀ at x = a .
  4. DE_x increases linearly in magnitude for |x| < R , decreases in magnitude as 1 x^2 for R < |x| < a , and exhibi

Correct answer

D. E_x increases linearly in magnitude for |x| < R , decreases in magnitude as 1 x^2 for R < |x| < a , and exhibi

Step-by-step solution

The net electric field E_x is the superposition of the fields from the two sheets and the solid sphere. 1. Field due to the two sheets: For -a For x > a , both sheets produce a field in the +x direction: E_ sheets = 2 ₀ + 2 ₀ = ₀ . For x Thus, the field due to the sheets is zero between them and jumps discontinuously by ₀ at x = a and x = -a . 2. Field due to the solid sphere: Inside the sphere ( |x| Outside the sphere ( |x| > R ), the field is E_ sphere = R^3 3 ₀ x^2 sgn (x) , which decreases in magnitude as 1 x^2

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