Magnitude and Direction of E-field at a Point Due to a Charged Rod

[MurdocJensen]

Homework Statement

There is a non-conducting charged rod with length L=.0815(m) and linear charge density λ=-5.9x10^-14(C/m). The rod is placed parallel to and on the x-axis, and at a distance a=.12(m) from the right-most end of the rod is point P. Calculate the magnitude and direction of E (electric-field) at point P due to the charged rod.

Homework Equations

E = [1/(4$\pi$$\epsilon$₀)][q/r²]

The Attempt at a Solution

First, we know we are dealing strictly with x-components of the E-field due to the charged rod because the x-axis runs through the length of the rod and point P is on the x-axis. The equation for E becomes E = [1/(4$\pi$$\epsilon$₀)][q/r²]cos($\theta$). At the same time we note that the angle that any element of the rod makes with point P is $\theta$=0, so the cos($\theta$) term is 1.

Because the charge is distributed throughout the rod uniformly (as opposed to being concentrated at a single point), we need to find the contribution for each charge element dq of the rod. I do this by changing q in my equation to dq.

At this point I have two variables, dq and r (E = [1/(4$\pi$$\epsilon$₀)][dq/r²]). Here's where my trouble begins. Since I'm not utilizing iterated integrals (don't mind you guys giving me tips on how to do so) I want to get all variables in terms of just one, namely dq=(λ)(dx) and r=.12(m)+(.0815(m)-x). Am I on the right path?

 

[SammyS]

It looks like a good start.

 

[MurdocJensen]

Alright. I was able to check the answer with a friend and I got it right. Thanks for your words.

Would an iterated integral be an easier way to approach this? Wouldn't I just keep my variables as is and just append their differentials to the end of the function?

 

[SammyS]

I don't see why you would use an iterated integral for this problem.

 

[rwisz]

Your approach to finding the magnitude and direction of the E-field at point P due to the charged rod is correct. However, there are a few things that could be clarified.

First, the equation you have written for the E-field is not quite correct. It should be E = [1/(4\pi\epsilon0)] (dq/r^2) * cos(theta), where dq is the infinitesimal charge element, r is the distance from the charge element to point P, and theta is the angle between the direction of the E-field and the line connecting the charge element to point P. This equation is derived from Coulomb's law, where we consider the contribution of each infinitesimal charge element to the total E-field at point P.

Next, in order to find the contribution of each charge element, you correctly noted that we need to express dq and r in terms of one variable. In this case, we can use the linear charge density lambda (λ) to express dq as dq = λ*dx, where dx is the infinitesimal length element along the rod. Similarly, we can express r as r = (0.12 + 0.0815 - x), where 0.12 is the distance from the right end of the rod to point P, 0.0815 is the length of the rod, and x is the distance from the right end of the rod to the infinitesimal charge element.

Finally, to find the total E-field at point P, we need to integrate the contributions from each infinitesimal charge element along the entire length of the rod. This can be done using the equation you derived, E = [1/(4\pi\epsilon0)] (dq/r^2) * cos(theta), with dq and r expressed in terms of x as described above. The integral will give you the magnitude of the E-field at point P, and you can use the direction of the E-field at each infinitesimal charge element to determine the overall direction of the E-field at point P.

In conclusion, your approach is correct, but it will require integration to find the total E-field at point P. I hope this helps clarify the process. Good luck with your calculations!