Three infinitely long thin charged sheets are placed parallel to the y-z plane at x = -d , x = 0 , and x = d . Their surface charge densities are 2 , 4 , and k respectively. If the electric field at the point x = d/2 is zero, the value of k is _______.

Step-by-step solution

Electric field at a point due to an infinite sheet of charge density is 2 ₀ directed away from the sheet if positive, and towards it if negative. At the point x = d/2 : The sheet at x = -d produces a field E ₁ = 2 2 ₀ i = ₀ i . The sheet at x = 0 produces a field E ₂ = 4 2 ₀ i = 2 ₀ i . The sheet at x = d produces a field E ₃ = - k 2 ₀ i . The net electric field is zero: E _ net = E ₁ + E ₂ + E ₃ = 0 ₀ i + 2 ₀ i - k 2 ₀ i = 0 3 - k 2 = 0 k = 6 . Answer: 6