Electric field of a ring:integrate over dq or ds?

[Tiago3434]

Hi people, I was reading on my textbook on electromagnetism (Halliday) about using integral to field the electric field of a charged ring (1D) at a point P located on an axis perpendicular to the ring's plane. The ring is uniformly charged. The book (and my professor) both breakdown each element of charge dq into its linear charge density times an element of length, or λds. I'm ok with the math from that point on, but I noticed that you can get the same result just integrating over dq, given that the integral of dq is q. Is my idea wrong, or is there a reason for adding that extra step?
Thanks!

 

[jtbell]

Suppose the charge density were not uniform. Integrating over s, you would let λ be some function of s, λ(s). Then your integral would contain λ(s)ds. How would you integrate directly over dq in that case?

 

[Tiago3434]

oh, ok I got you point. Thanks, jtbell

 

[Tiago3434]

Sorry if the question is stupid, but thinking a bit further about this, wouldn't the overall charge just add up to q again (even if all the dq weren't equal)?

 

[Tiago3434]

Or (sorry if this is a calculus question) can you only integrate over a differential dx if dx is constant? Is that the case?

 

[pixel]

Sorry if the question is stupid, but thinking a bit further about this, wouldn't the overall charge just add up to q again (even if all the dq weren't equal)?

The integral of λ(s)ds would be q, the total charge. But you are trying to calculate the electric field, E, at a point, P, on the axis, not the total charge. λ(s) can't be taken out of the integral over s in that case.

 

[Tiago3434]

I'm sorry but I don't get it. Wouldn't the answer just end up being the same, as dq=λ(s)ds?

 

[Tiago3434]

Like, if dq=λds, I don't understand how come the integrals ∫dq and ∫λds could end up being different.

 

[NFuller]

Like, if dq=λds, I don't understand how come the integrals ∫dq and ∫λds could end up being different.

Those integrals are not different but those are not the integrals for the electric field. The integral for the electric field also contains a directional element $\hat{r}$ which may weight the electric field more in one direction than the other.

In general
$$\mathbf{E}=\frac{1}{4\pi\epsilon_{0}}\int\frac{dq}{r^{2}}\hat{r}$$

As an example, let's assume you are looking for $\mathbf{E}$ at the center of the ring, then $\hat{r}=-\text{cos}\theta\hat{x}-\text{sin}\theta\hat{y}$. If the ring has a radius of $R$ then
$$\mathbf{E}=\frac{-1}{4\pi\epsilon_{0}}\int\frac{dq}{R^{2}}(\text{cos}\theta\hat{x}+\text{sin}\theta\hat{y})$$
Now let's assume the ring has some varying linear charge density $\lambda=\alpha\text{cos}\theta$ then
$$dq=\lambda Rd\theta=\alpha\text{cos}\theta\;Rd\theta$$
The electric field is now
$$\mathbf{E}=\frac{-1}{4\pi\epsilon_{0}}\int_{0}^{2\pi}\frac{\alpha\text{cos}\theta\;Rd\theta}{R^{2}}(\text{cos}\theta\hat{x}+\text{sin}\theta\hat{y})$$
$$\mathbf{E}=\frac{-\alpha}{4\pi\epsilon_{0}R}\left[\int_{0}^{2\pi}\text{cos}^{2}\theta\;d\theta\hat{x}+\int_{0}^{2\pi}\text{cos}\theta\text{sin}\theta\;d\theta\hat{y})\right]$$
$$\mathbf{E}=\frac{-\alpha}{4\pi\epsilon_{0}R}\left[\pi \hat{x}+0\hat{y})\right]=\frac{-\alpha}{4\epsilon_{0}R}\hat{x}$$
Now if we consider the case of constant charge density $\lambda$
$$\mathbf{E}=\frac{-1}{4\pi\epsilon_{0}}\int_{0}^{2\pi}\frac{\lambda Rd\theta}{R^{2}}(\text{cos}\theta\hat{x}+\text{sin}\theta\hat{y})$$
$$\mathbf{E}=\frac{-\lambda}{4\pi\epsilon_{0}R}\left[\int_{0}^{2\pi}\text{cos}\theta\;d\theta\hat{x}+\int_{0}^{2\pi}\text{sin}\theta\;d\theta\hat{y})\right]$$
$$\mathbf{E}=\frac{-\lambda}{4\pi\epsilon_{0}R}\left[0 \hat{x}+0\hat{y})\right]=0$$
These two answers are different because including the direction $\hat{r}$ selects which parts of the charge density will add together in some direction and which will cancel out in another direction.