Question

An infinitely long thin wire carries a uniform charge per unit length `q`. Find the electric field at a distance of `R` units from the wire?

Answer 1

Electric field is `q/(4epsilon_0R)`, where `q` is the uniform charge per unit length and `epsilon_0` is electrical permittivity of free space

Explanation:

Consider the infinitely long thin wire to carry a uniform charge per unit length `q`,

The field created by a small portion `dx` of it at a distance `R` will be

`1/(4piepsilon_0)xx(qdx)/sqrt(x^2+R^2)`, where `epsilon_0` is electrical permittivity of free space

and total field created by thin wire will be

`1/(4piepsilon_0)int_(-oo)^(oo)(qdx)/sqrt(x^2+R^2)`

= `q/(4piepsilon_0)int_(-oo)^(oo)(dx)/sqrt(x^2+R^2)`

= `q/(4piepsilon_0)1/R|tan^(-1)(x/R)|_(-oo)^(oo)`

= `q/(4piepsilon_0R)(pi/2-(-pi/2))`

= `q/(4epsilon_0R)`