Electric Field Intensity Due to Two Infinite Parallel Series of Different Charges

When two infinite parallel series of different charges are placed opposite to each other, the electric field intensity in the region between the two plates is given by:

E = σ1/2ε0 - σ2/2ε0

where σ1 and σ2 are the surface charge densities of the two plates, and ε0 is the permittivity of free space.

Derivation

1. Define the Problem

Consider two infinite parallel series of plates, one with a positive charge density σ1 and the other with a negative charge density σ2. Both plates are separated by a distance d.

2. Determine the Electric Field due to each Plate

The electric field due to the positive plate is given by:

E1 = σ1/2ε0

The electric field due to the negative plate is given by:

E2 = -σ2/2ε0

Note that the negative sign indicates that the electric field points in the opposite direction to that of the positive plate.

3. Apply the Superposition Principle

Since the electric field is a vector quantity, the total electric field in the region between the plates is the vector sum of the electric fields due to each plate. By the principle of superposition, we can add the two electric fields together to get:

E = E1 + E2

Substituting the expressions for E1 and E2, we get:

E = σ1/2ε0 - σ2/2ε0

This expression gives the electric field intensity in the region between the two plates.

Conclusion

The electric field intensity due to two infinite parallel series of different charges can be determined by adding the electric fields due to each plate using the principle of superposition. The resulting equation is E = σ1/2ε0 - σ2/2ε0.