Two identical infinite non-conducting sheets having equal positive surface charge densities sigma\sigma are kept parallel to each other as shown in the Figure below. Determine the electric field at a point in (a) region AA on the left of the sheet 1, (b) region BB between the sheets and (iii) region CC on the right of the sheets.
Answer:
We need to consider the superposition principle, which states that the net electric field is the vector sum of electric fields produced by each charge distribution. For an infinite sheet with a uniform positive surface charge density sigma\sigma, the electric field at any point in space is perpendicular to the surface of the sheet and has a magnitude of E=(sigma)/(2epsilon_(0))E = \frac{\sigma}{2\epsilon_0}, according to Gauss’s law.
Let’s consider the situation described, with two identical infinite non-conducting sheets parallel to each other, each with a surface charge density of sigma\sigma.
(a) Region A (to the left of sheet 1):
Since sheet 1 has a positive surface charge density sigma\sigma, it creates an electric field directed away from itself. By symmetry, the electric field at any point to the left of sheet 1 will be directed to the right, perpendicular to the sheet, with a magnitude of (sigma)/(2epsilon_(0))\frac{\sigma}{2\epsilon_0}.
The electric field contribution from sheet 2 to the left of sheet 1 is exactly canceled by the field due to sheet 1, because the field from sheet 2 at that point is directed to the left, while the field from sheet 1 is to the right. Thus, the net electric field in region A is solely due to sheet 1 and is given by:
In this region, both sheets contribute to the electric field. The electric field due to sheet 1 is directed to the right, and the electric field due to sheet 2 is also directed to the right, away from sheet 2. Since the sheets are identical, each contributes an electric field with a magnitude of (sigma)/(2epsilon_(0))\frac{\sigma}{2\epsilon_0}. These fields add up due to superposition.
The net electric field in region B is the sum of the fields due to both sheets:
For region C, the analysis is similar to region A. The electric field due to sheet 1 is canceled by the field due to sheet 2, and the net electric field in region C is solely due to sheet 2.
The electric field in region C, similar to region A, is directed away from sheet 2, to the right, with a magnitude of: