Solving Electric Potential & Field of Continuous Charge Rod: Q&A

[oooride]

A rod with a continuous charge extending from x=2.00m to x=3.00m has a charge density lambda=3x^2.

a.)What is the charge on the rod?
b.)What is the Electric potential of the rod at the origin?
c.)What is the Electric field at the origin.


This was a test question that I totally messed up...

How do I find the charge of the rod? Everything I did wrong stemmed off that first part of the question.

For part a I totally messed up, I'm sure it was extremely easy, but I ended up integrating from 2 to 3 dq/r^2 only to realize with no time left that I'm finding the E field of rod, not the charge...[b(] Thinking about it now, should I have treated it as a guassian surface and found the charge by q_in / epsilon knot ?


When finding the electric potential I used V = Kq/r because I figured since it was the origin it was defined as V=0 and R_a = infinity.

When finding the E field at the origin I again treated the origin as a point and used E = kq/r^2 but then after the test realized I probably should have used E=int Kdq/r^2 and integrated from 0 to 3..
Should I of?

Any help is greatly appreciated
Thanks

 

[cookiemonster]

Be careful with units. You really need to mention what units the charge density is given in. I'll assume C/m.

a)$$Q = \int_a^b \lambda\,dx = \int_2^3 3x^2\,dx$$

b)$$V = \frac{1}{4\pi\epsilon_0}\int_a^b \frac{\lambda}{r}\,dx = \frac{1}{4\pi\epsilon_0}\int_2^3 \frac{3x^2}{x}\,dx$$

you really should have stated the y-location of the rod. I took it to be 0.

c)$$E = \frac{1}{4\pi\epsilon_0}\int_a^b \frac{\lambda}{r^2}\,dx = \frac{1}{4\pi\epsilon_0}\int_2^3 \frac{3x^2}{x^2}\,dx$$
obviously in the -x direction.

cookiemonster

 

[Doc Al]

Originally posted by oooride
How do I find the charge of the rod? Everything I did wrong stemmed off that first part of the question.

By integrating the charge density (λ= 3x^2) over the interval x = (2,3).

... Thinking about it now, should I have treated it as a guassian surface and found the charge by q_in / epsilon knot ?

Gauss's law won't help.

When finding the electric potential I used V = Kq/r because I figured since it was the origin it was defined as V=0 and R_a = infinity.

I'm not sure what you're talking about. V = kq/r is the standard form for the potential at distance "r" from the charge "q". (Yes, that defines the potential as zero at r = infinity.) So if you integrated kq/x over the interval x = (2,3), where q = λdx, you got it.

When finding the E field at the origin I again treated the origin as a point and used E = kq/r^2 but then after the test realized I probably should have used E=int Kdq/r^2 and integrated from 0 to 3..
Should I of?

You should have integrated kq/x^2, where q = λdx, but over the interval x = (2,3), not (0,3). There is no charge in the interval x = (0,2), so what would you be integrating?

Edit: Hey, cookiemonster beat me to it!

 

[cookiemonster]

I know I always just wanted to see the math after a test, so I kind of forewent the explanations. Guess now he's got both!

cookiemonster

 

[Doc Al]

Originally posted by cookiemonster
I know I always just wanted to see the math after a test, so I kind of forewent the explanations. Guess now he's got both!

And I was too lazy to write the equations properly using Latex like you did. We make a good tag team.

 

[oooride]

Hmmm, well it looks like I got it wrong, which isn't good on a three question test.

Yeah I understand what was done, but got confused when it said the origin was involved.

Thanks for all the help.