Two infinite long straight wires having surface charge density +λ and -λ are kept parallel to each other at a small separation distance d. Find the electric field intensity at the mid point between the wires.

CONCEPT:

Electric Field due to an infinitely long straight uniformly charged wire:

• The intensity of the electric field due to a uniformly charged infinite wire of radius R at a distance r (> R) from its axis is given as,

\(⇒ E=\frac{λ}{2\pi r\epsilon_{o}}\)

Where λ = surface charge density

EXPLANATION:

• The midpoint between the wires will be at an equal distance from both the wires, so the magnitude of the electric field is the same at the midpoint due to the charge on both the wires.
• The magnitude of the electric field intensity at the midpoint due to any of the wire is given as, (\(r=\frac{d}{2}\))

\(⇒ E'=\frac{2λ}{2\pi \epsilon_{o}d}\)

\(⇒ E'=\frac{λ}{\pi \epsilon_{o}d}\)     -----(1)

• From the diagram, it is clear that the direction of the electric field intensity due to both the wires will be the same at the midpoint.
• So the net electric field intensity at the midpoint is given as,

⇒ E = E' + E'

⇒ E = 2E'

\(⇒ E'=\frac{2λ}{\pi \epsilon_{o}d}\)

• Hence, option 1 is correct.