In this problem we will calculate the electric field of a li

In this problem, we will calculate the electric field of a line charge. The line charge is aligned along the x-axis starting at the origin and having a length L. The line has a linear charge density A. We want to find the electric field at a point P on the z-axis. The point P is located at the point (d, 0) where d> L. (a) List all of the given parameters. (b) What is the dimensionality of a line of charge? What is the electric field of a small charge dq which is one dimension less than our line? (c) What is the charge dq in terms of the given parameters? (d) Wrhat is the distance between an arbitrary point on our line charge to the point P in terms of given parameters? (e) What is the unit vector r in terms of given parameters? (f)What is the expression for the electric field of the line in integral form? This means plug in your answers to parts (c), (d), and (e) into your answer for part (b) but don\'t do the integral yet. (g) Rewrite this expression with all the constants pulled out of the integral. Include the bounds of the integral if you have not done so already.

Solution

We need to determine the electric field due to a line charge kept along the x axis and of length L. The point under consideration is d and the charge density we have is . Plus, we need to keep into consideration the following points:

1.) The electric field due to a charge q and at distance r from the charge is given as kq/r^2

2.) Electric field is a vector quantity which points away from the charge in case it is positive and towards the charge in case it is negative.

We would now solve the problems as follows:

a.) The parameters we have: charge density =  

length of rod = L

Distance of point under consideration from the right end = d - L

Distnace of point P from left end = d

b.) For a line charge, the distribution is linear in nature and extends along one dimension; in the given case, along the x axis.

Electric field due to a small charge dq would be Kdq/r^2

c.) Any small charge dq would be of length, dx, along the rod. Also, we have the charge density as  

Hence, the magnitude of the charge dq would be: dx

d.) The distance of any arbitrary point, say at x along the length, from the point P would be: d - x

e.) The unit vector r for the electric field would be along the x axis.

f.) Now, for any distance x and small charge dq = dx, the electric field at the point P would be:

dE = K dx / (d-x)^2

Now, as the direction of the electric field for all such small charges would be along the x axis, the net field due to the given line charge would be along the x axis and the magnitude would be integration of all the charges.

That is, E =  dE =  K dx / (d-x)^2

g.) We rearrange the integral as required:

E = K dx / (d-x)^2 [The limits of the integral would be from 0 to L]

h.) The given integral is of the form dx/x^2, hence the integral would -1/x. Using d-x in place of x we would get integral would be of the form 1 / (d-x)

That is, E = K [1 /(d-L) - 1/d] = KL / d(d-L) and the direction would along +x axis