Calculating the Electric Field of a Charged Rod: Is There an Easier Way?

[tony873004]

Can someone tell me if I did this correctly, or if there's an easier way? Thanks!

A uniformly charged rod of length $$\ell$$ with a charge density $$\lambda$$ lies along the x-axis, with its midpoint at the origin. Find the electric field at a point on the x-axis, with $$x > \ell /2$$

The electric field of a differential element is
$$\overrightarrow {dE} = \frac{{kdQ}}{{r^2 }}$$

The charge of a differential element is
$$dQ = \lambda \,dx$$

Let P be a point on the x-axis to the right of the rod.
$$\overrightarrow E = \int\limits_{ - \ell /2}^{\ell /2} {\frac{{k\lambda \,dx}}{{r^2 }}}$$

The distance r from P to a differential element x is P-x
Pull out the constants.
$$\begin{array}{l}<br /> \overrightarrow E = k\lambda \int\limits_{ - \ell /2}^{\ell /2} {\frac{{\,1}}{{\left( {P - x} \right)^2 }}dx} \\ <br /> \\ <br /> \overrightarrow E = k\lambda \left[ {\frac{1}{{P - x}}} \right]_{ - \ell /2}^{\ell /2} = k\lambda \left( {\left( {\frac{1}{{P - \ell /2}}} \right) - \left( {\frac{1}{{P - \left( { - \ell /2} \right)}}} \right)} \right),\, - {\rm{\hat i}} \\ <br /> \\ <br /> \overrightarrow E = k\lambda \left( {\left( {\frac{1}{{P - \ell /2}}} \right) - \left( {\frac{1}{{P + \ell /2}}} \right)} \right) \\ <br /> \end{array}$$

 

[Google_Spider]

IMO, its correct.

 

[tony873004]

I just realized that the answer is in the back of the book. They give:
$$\frac{{k\lambda \ell }}{{x_0^2 - \frac{1}{4}l^2 }}{\rm{\hat i}}$$

Where did I go wrong? I'm guessing that their x₀ is what I was calling P. But the answers are not the same. I don't believe they are the same formula just written differently.

Also, the direction given is i-hat. But since the problem never states what the charge of the rod is, how do I know that it is not negative i-hat?

 

[Gear300]

Since x > P, then r should be (x - P). I'm able to get the answer without the L in the numerator. Dont know where that's coming from.

 

[tony873004]

x > L/2, meaning the point is to the right of the rod. I choose P as the point so x could be my integration variable. So, for example if the rod's length were 10, it would span from x= -5 to 5. And if P was 15, then the distance from P to each element in the rod would be P-x. For example P is 15-5=10 units from the right of the rod.

I just verified using my calculator that my formula and the book's formula are the same. I just made up some numbers and plugged them in: k=12, lambda=13, L=14, P=15

12*13*((1/(15-14/2))-(1/(15+14/2))) = 12.4090909090909
12*13*14/(15^2-14^2/4) = 12.4090909090909

The answers are the same. So its just a matter of algebra to arrive at their formula from my formula. But I don't know how. I'm wondering if I care. Is their formula any more correct than mine?