Question

In this question, we are going to solve the "Long line of charge" problem by using the Gauss' Law Imagine a very long uniformly charged straight thin rod, charge +λ C/m per unit length. Determine the electric field at any point P a distance x from the the rod X.

Pick a random point anywhere on the rod (far from the edges) and consider another infinitesimal piece of the same length the same distance away from the random point you picked as the first piece and the electric field that it produces. How would the electric fields produced by these two pieces add? Would they enhance each other in certain directions? Would they cancel each other in certain directions?

Answer 1

3 M dLE 0 d E SmO

dq = small charge on the small length = $\lambda$ dy

small electric field by the small charge at P is given as

dE = k dq/(x² + y²) = k $\lambda$ dy /(x² + y²)

the component of electric field along Y-direction , being equal in magnitude and opposite in direction from the small charges on both side , cancel out , hence the net electric field is along X-direction which is given as

E = $\int_{-\infty }^{\infty }$ dE_x = $\int_{-\infty }^{\infty }$ k dq/(x² + y²) Cos$\theta$ =  $\int_{-\infty }^{\infty }$ k $\lambda$ dy x/(x² + y²)^3/2

E = 2k$\lambda$/x

radially outward(perpendicular to rod) and uniform

a cylinder concetric with rod