Electric field of a semi circle ring

[hitemup]

Homework Statement

Suppose the charge Q on the ring of Fig. 28 was all distributed uniformly on only the upper half of the ring, and no charge was on the lower half. Determine the electric field $\vec {E}$ at P.

Homework Equations

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$$E = k\frac {q} {r^2}$$

The Attempt at a Solution

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Starting with the charge density,

$$\lambda = \frac{Q}{\pi a} = \frac{dq}{dl}$$
electric field for dq
$$dE = k\frac{dq}{x^2+a^2}$$
Let's find its x component
$$dE_x = dEcos\theta = k\frac{dq}{x^2+a^2}cos\theta = kx\frac{dq}{(x^2+a^2)^{3/2}}$$
$$dE_x = \frac{k\lambda x}{(x^2+a^2)^{3/2}}dl$$

Wıth the limits being $0$ and $\pi a$

$$E_x = \frac{k\lambda x \pi a}{(x^2+a^2)^{3/2}} = \frac{kqx}{(x^2+a^2)^{3/2}}$$
This is correct according to my textbook.
However, I end up with a wrong y component when I apply the same logic for the vertical.
$$E_y = -\frac{k\lambda a \pi a}{(x^2+a^2)^{3/2}} = -\frac{kqa}{(x^2+a^2)^{3/2}}$$

But the correct answer for the vertical is
$$E_y = -\frac{kq2a/\pi}{(x^2+a^2)^{3/2}}$$

Having searched the internet for this question, I found that the solution for the y-axis includes an arc length element, which is something like this.
$$\lambda = \frac {dq} {a d\theta}$$
Then I suppose, it integrates from $0$ to $\pi$ for a semi circle.

So my question is, how does that differ if one uses $d\theta$ or $dl$ for integration? Also, why did I get the x component correct, even though the fact that I used $dl$?

 

[haruspex]

So my question is, how does that differ if one uses $d\theta$ or $dl$ for integration? Also, why did I get the x component correct, even though the fact that I used $dl$?

For the x component, the contribution of each element was independent of theta. Not so for the y component.

 

[TSny]

You didn't show the details of your work for the y component.

Note that using sinθ instead of cosθ will give you the component of $d\vec{E}$ that is perpendicular to the x axis. But this perpendicular component is not parallel to the y-axis (in general).

 

[hitemup]

You didn't show the details of your work for the y component.

Note that using sinθ instead of cosθ will give you the component of $d\vec{E}$ that is perpendicular to the x axis. But this perpendicular component is not parallel to the y-axis (in general).

It is much like the work I've done for the x component, but let me write it any way.

$$dE_y = dEsin\theta = k\frac{dq}{x^2+a^2}sin\theta = ka\frac{dq}{(x^2+a^2)^{3/2}}$$
$$dE_y = \frac{k\lambda a}{(x^2+a^2)^{3/2}}dl$$
$0$ to $\pi$
$$E_y = \frac{k\lambda a \pi a}{(x^2+a^2)^{3/2}} = \frac{kqa}{(x^2+a^2)^{3/2}}$$

As you say, I'm probably missing something geometrical, but I can't see how dE*sin(theta) is not parallel to the y-axis in this particular situation. If its cosine component is parallel to the x axis, shouldn't its sine component be parallel to the y axis?

 

[TSny]

As you say, I'm probably missing something geometrical, but I can't see how dE*sin(theta) is not parallel to the y-axis in this particular situation.

Consider an infinitesimal charge element located at one end of the semicircular ring. In what direction does the perpendicular component dE*sin(theta) point? (I'm imagining the lower half of the ring chopped off and removed so the semicircle that is left has two ends.)

 

[haruspex]

If its cosine component is parallel to the x axis, shouldn't its sine component be parallel to the y axis?

There are three dimensions here. If the cosine gives the component parallel to the x-axis then the sine gives the component orthogonal to the x axis, but that does not make it always parallel to the y axis.

 

[hitemup]

Consider an infinitesimal charge element located at one end of the semicircular ring. In what direction does the perpendicular component dE*sin(theta) point? (I'm imagining the lower half of the ring chopped off and removed so the semicircle that is left has two ends.)

I think it doesn't have a y component because the ends of the circle is at the same height with point P, so I believe there is no y since there is no elevation.

 

[hitemup]

There are three dimensions here. If the cosine gives the component parallel to the x-axis then the sine gives the component orthogonal to the x axis, but that does not make it always parallel to the y axis.

Yes, now I got it. I've been playing with my hands for the last 10 minutes, making a circle with one and trying to understand what is going on. I see that all sine components cannot be added together because they do not point the same direction. Am I right?

 

[TSny]

Yes. Good.

 

[hitemup]

Yes. Good.

I understand that we are in a 3d coordinate system, and sine component is perpendicular to x axis. Also, that perpendicular component is not always parallel to the y axis. So how can I proceed to the y components knowing these?
Thank you so much for your help by the way.

 

[TSny]

Pick an element of charge and imagine a line drawn from the origin to the element. This line will make some angle φ to the y axis. Imagine a plane that contains this line and the x axis. Can you see that the electric field produced by the element of charge will lie in this plane? So, the component of the electric field that is perpendicular to the x-axis will also lie in this plane. See if that helps in finding the y component of the field.

 

[hitemup]

Pick an element of charge and imagine a line drawn from the origin to the element. This line will make some angle φ to the y axis. Imagine a plane that contains this line and the x axis. Can you see that the electric field produced by the element of charge will lie in this plane? So, the component of the electric field that is perpendicular to the x-axis will also lie in this plane. See if that helps in finding the y component of the field.

If my setup looks correct, then I would like to ask how to find a relationship between beta and theta.

 

[TSny]

You won't need a relationship between those two angles. Notice that θ is a fixed angle while β varies to cover all the charge elements. Your diagram looks good. How would you project $dE_\perp$ onto the the vertical dotted line?

 

[hitemup]

You won't need a relationship between those two angles. Notice that θ is a fixed angle while β varies to cover all the charge elements. Your diagram looks good. How would you project $dE_\perp$ onto the the vertical dotted line?

Multiplying it with $cos \beta$?
If I am right, then I need limits for the integral. Would it be 2*[from pi/2 to zero], or 2*[zero to pi/2]?

 

[TSny]

I would use twice the integral from 0 to pi/2.

 

[hitemup]

I would use twice the integral from 0 to pi/2.

Thank you so much, I've managed to get the correct answer.
Just one more thing. Should the final result's sign always obey the direction of the vector?

 

[TSny]

Yes. If you are asked for E_y, then the answer should be negative. If you are asked for the magnitude of the y component, then that generally means just the absolute value.