Question

In: Physics

Two infinitely long parallel rods carrying uniform charge density lambda are at a distance d from...

Two infinitely long parallel rods carrying uniform charge density lambda are at a distance d from each other. Find the electric force on a particle of charge q_* located a distance d directly left of one of the rods. (Derive the formula using Gauss' Law)

Expert Solution

Consider a Gaussian cylindrical surface of radius "d" and length "L" for rod closer to the particle and of radius "2d" and length "L" for rod farther to the particle :

For the closer rod :

A = Area of the Gaussian surface = 2dL

Q_enclosed = amount of charge enclosed = L

E_closer = Electric field by closer rod on the particle

Using Gauss's law

E_closer A = Q_enclosed /

E_closer (2dL) = L/

E_closer = 2 /(4d) eq-1

For the farther rod :

A = Area of the Gaussian surface = 2(2d)L

Q_enclosed = amount of charge enclosed = L

E_farther = Electric field by farther rod on the particle

Using Gauss's law

E_farther A = Q_enclosed /

E_farther (4dL) = L/

E_farther = /(4d) eq-2

Using eq-1 and eq-2

E = Total electric field at the location of charge = E_closer + E_farther

E = 2 /(4d) + /(4d)

E = 3 /(4d)

Force on the particle is given as

F = q E

F = 3 q/(4d)

genius_generous answered 8 hours ago

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