Consider a uniform line change of length 2l placed along x axis with center at the ...

Consider a uniform line change of length 2l placed along x axis with center at the origin and line charge density 2= kx² where is a constant. Find the electric field at a distance 2 about the midpoint.

Solution

The electric field at a point along the x-axis due to a line charge can be found using Coulomb's Law. The electric field at a point along the x-axis due to a section of the line charge of length dx at a distance x from the origin is given by:

dE = (kx^2 * dx) / (x^2 + 4)^(3/2)

Integrating this over the length of the line charge, we get the total electric field at the point (2, 0):

E = (2/4^(3/2)) * ∫[-l,l] (kx^2) / (x^2 + 4)^(3/2) dx

Evaluating this definite integral, we find the electric field at the point (2, 0) to be:

E = (k * 2 * 2^(3/2)) / 3

So the electric field at a distance of 2 units from the midpoint of the line charge of length 2l is (k * 2 * 2^(3/2)) / 3.

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