Two parallel infinite line charges with linear charge densities +λC/m and −λC/m are placed at a distance of 2R in free space. What is the electric field mid-way between the two line charges?

• A. zero
• B. 2λ / πε₀R
• C. λ / πε₀R
• D. 1 / 2πε₀R

To determine the electric field midway between two parallel infinite line charges with charge densities +λ C/m and −λ C/m, we can analyze the contributions from each line charge.

Electric Field from a Line Charge

The electric field (E) due to an infinite line charge is given by the formula:

• E = λ / (2πε₀r)

where λ is the linear charge density, ε₀ is the permittivity of free space, and r is the distance from the line charge.

Calculating the Fields

In this scenario, the distance from each line charge to the midpoint is R (since the total distance is 2R).

For the positive line charge (+λ), the electric field at the midpoint points away from the charge:

• E₁ = λ / (2πε₀R)

For the negative line charge (−λ), the electric field at the midpoint points towards the charge:

• E₂ = λ / (2πε₀R)

Net Electric Field

Since both fields have the same magnitude but opposite directions, they will cancel each other out:

• E_net = E₁ - E₂ = (λ / (2πε₀R)) - (λ / (2πε₀R)) = 0

Final Answer

The electric field midway between the two line charges is zero.