Three infinitely long charged thin sheets are placed as shown in figure. The magnitude of electric field at the point P is (x σ)/ε₀. The value of x is _________ (all quantities are measured in SI units).

Solution

Physics sir · 3 steps

P sits between the sheets at x = a and x = 3a, so check which way each sheet pushes there, then add.

1. One sheet. By Gauss's law, an infinite sheet with charge density σ gives a uniform field of magnitude σ/2ε₀. It is the same at every distance. It points away from a positive sheet and towards a negative sheet.

   Sheet	Where P is	Magnitude at P	Direction at P
   -σ at x = -a	to its right	σ/2ε₀	towards the sheet, -x
   -2σ at x = a	to its right	2σ/2ε₀ = σ/ε₀	towards the sheet, -x
   +σ at x = 3a	to its left	σ/2ε₀	away from the sheet, -x

2. Add them. All three fields point along -x, so their magnitudes add directly.

   E_P = σ/2ε₀ + σ/ε₀ + σ/2ε₀ = 2σ/ε₀

3. So. The field at P has magnitude 2σ/ε₀ along -x, so x = 2.

The common slip: 1. Adding the charge densities with their signs, (-σ - 2σ + σ)/2ε₀, gives a magnitude of σ/ε₀, so 1. The +σ sheet lies to the right of P, so its field at P points along -x, the same way as the other two.

The idea to keep. For parallel sheets, do not add the charge densities with their signs. Find the direction of each sheet's field at the point first, using which side of the sheet the point is on. Then add.

Answer from the NTA final answer key. Solution by PrepWiser.