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**1. Homework Statement**

A thin rod of length 2ℓ has a linear charge density that is λ

_{0}at the left end but decreases linearly with distance going from left to right in such a way that the charge on the entire rod is zero.

What is the magnitude of the electric field along the rod's axis at a position P that is a distance d>ℓ to the right of the center of the rod? Use the notation l for the variable ℓ.

Express your answer in terms of the variables l, λ

_{0}, d, and Coulomb's constant k.

**2. Homework Equations**

Electric field = kq/d

^{2}

λ=q/l

**3. The Attempt at a Solution**

I am completely lost on this one, although I have some parts down I think.

I know that I will need to integrate and should be using a small section of rod dl for charge dq.

One end of the rod has charge density of λ

_{0}, so the other end of the rod must have charge -λ

_{0}in order for the entire rod to have a zero net charge. I also think that you would integrate from 0 to 2l. I have tried several different setups but do not feel confident on any of them and could not find any guidance from a similar problem on the Internet or my textbook.

Because the charge decreases linearly, I was thinking that dq = λ

_{0}- ax (if one end of the rod is placed at the origin), but that just added another variable which confused me further.

At first I had tried to consider the rod as a dipole, since one end was positive and one was negative, but it seems wrong that way since there is charge throughout and not just at both ends.

Any help with this is greatly appreciated, thanks.