Line of charge, Feild on the origin

[K3nt70]

Homework Statement

A line of charge with a uniform density of 35.0 nC/m lies along the line y = -12.6 cm, between the points with coordinates x = 0 and x = 44.2 cm. Calculate the electric field it creates at the origin. Find the x- and y-components of the field.

Homework Equations

I believe i have to use an integrated formula but I am not sure which one.
http://img385.imageshack.us/img385/8756/66883216ju0.png

The Attempt at a Solution

I'm not sure if i have the problem straight in my head, but if i do, then i don't know how to approach it. I know that i have to get the x and y components from the magnitude, and i know how to do that, but i don't really know how to find the magnitude.

 

[cryptoguy]

if this was a point charge, you would figure out the electric field by using E = kQ/r^2, right? So for a continuous charge distribution, dE = k*dQ/r^2, which you will have to integrate.

 

[K3nt70]

so after integration, the formula is:

E = k*Q/(x^2 + a^2) right?

 

[K3nt70]

no, that can't be right. If it were, that would mean the magnitude was only 659 N/C which is way to low. I must not be integrating properly?

 

[cryptoguy]

well no, that's actually more like the equation before integration (with dE and dQ). Once you get $$\int k \frac{dQ}{x^2 + a^2}$$ there's a problem because you're integrating with respect to Q (hence the dQ) but you want to integrate with respect to x. So replace dQ with $$\lambda dx$$. Now you're integrating x with respect to x.

 

[K3nt70]

so then that leaves me with E = $$\frac{-k\lambda}{x} - \frac{k\lambda}{a}$$
correct?

 

[cryptoguy]

hmm i don't think so, i don't remember off the top of my head what this integral would evaluate to but I'm relatively sure it involves a trig substitution.

 

[K3nt70]

Hm. Ok, ill have to put this one down for a bit and come back to it. Doing the integral is what is most confusing to me.

 

[Defennder]

You have to set up the differential electric fields in the x and y directions properly first. You can't just integrate with respect to Q because that doesn't take into account the geometrical setup of the problem.

Now, you can see by means of a diagram (draw it) that $$dE_y = \cos \theta dE$$, where $\theta$ ranges from 0 to some value and is basically the angle in between the y-axis and the imaginary radial line connecting the differential charged length as it sweeps from x=0 to x=44.2 cm.

So we have $$dE = \frac{\lambda dx}{4 \pi \varepsilon r(x)^2}$$. I wrote r(x) instead of r because r is not a constant. Find an expression for r, the length from the origin to the charged dx line segment. Now when you're done, plug in [itex]dE_y[/tex] into the formula for $dE$ above. You still have to express either cos theta in terms of x or all the x values in the dE expression as functions of theta before performing the integration. I recommend doing the former since it's easier to express cos theta in terms of x and some other constants. Once, you're done simply integrate over the length of the charged line.

The expression for $E_x$ can be found in a similar manner.

 

[K3nt70]

So doing the integral of:
$$dE = \frac{\lambda dx}{4 \pi \varepsilon r(x)^2}$$

yeilds:
$$E = \frac{-\lambda}{4 \pi \varepsilon x}$$

correct?

 

[Defennder]

You're supposed to be integrating to find either $$E_y \ \mbox{or} \ E_x$$. But more importantly you haven't found r(x) yet. r(x) as I wrote it means r as a function of x, not r*x, as you appear to have written it.

 

[K3nt70]

OK, so just that i have this correct.. i neet to integrate two equations one for Ey and one for Ex. Those would be:
dEx = dECos$$\theta$$
dEy = dESin$$\theta$$

This means that $$Ex = \frac{kx}{(x^2 + a^2)^\frac{3}{2}}Q$$
and Ey is the same thing only y values in place of x values?

 

[Defennder]

What is Q here? You're supposed to set up the integration in terms of $dE_x$, so how did you get that expression? REmember that the final answer for both Ex and Ey is a numerical value, it should not have x and y in it at all. I assume a here is 12.6cm.

 

[K3nt70]

Q is the charge on the rod. I used circular geometry and the lambda given to get the charge. Since radius is constant, its easy to find the circumfrence of what would be a circle. Then i used the ratio of 56:304 degrees to find the length of the rod. Length of rod multiplied by charge/length gives charge.

 

[Defennder]

You're not understanding the technique for doing such problems. First you have to find the differential electric field contribution due to each differential charged segment, then express it in terms of a variable and then integrate it over the geometrical parameters of the problem. I have given you the expression for $dE_y$ and told you to express $\cos \theta$ in terms of x so that the integration can be performed over the length of the charged rod. As as I have said earlier, the expression for $dE_x$ can be found in a similar manner.

And as for your reply, I think you've confused this thread and another thread of yours on the e-field due to a circular arc of charge. This one concerns a line, and the "radius", which I take to be the distance between the origin and the differential charged segment is clearly not constant.

 

[K3nt70]

Ok. So the geometrical parameters for this problem only range from 0 to 44.2 cm along the x-axis. this causes the hypotenuse to change. I believe this is what you meant by saying r(x) as in r as a function of x? I am thinking this would be $$r(x) = \frac{x}{\sqrt{x^2 + y^2}}$$ .

$$Cos\theta$$ expressed in terms of x is simply r(x) as i have it above, is it not?

If this is correct, then doing the integral still gives:

$$E = \frac{-\lambda}{4 \pi \varepsilon x}$$

which is to be evaluated from 44.2 to 0, right?
and yes i did get this problem confused with the other one for that last reply.

 

[Defennder]

r(x) is simply the distance from the differential charged segment $\lambda dx$ to the origin. Your expression is not correct. Just use pythagoras theorem to get it.

As for $\cos \theta$, which angle are you using here? You have to be consistent with your choice of angle when you perform the integration later. The one I used is the angle in between the y-axis and the line connecting the origin to the charged segment dx. You could use the other angle as well. Your current expression for cos theta uses the other angle.

The expression I wrote for $$dE_y$$ uses the angle I described earlier. And before doing the integration I suggest you post the expression for either $dE_y$ or $dE_x$ first, because your answers don't look look correct and I can't tell what's wrong.

 

[K3nt70]

r(x) is 46 cm.

The theta would be 15.9 deg.

The expression would be:

$$Ex = \frac{\lambda x^2}{4 \pi \varepsilon r(x)^3}$$
exaluated from 44.2cm to 0?

 

[Defennder]

I don't know how you got that, but that's not right. r(x) is a function of x, not the some numerical distance in itself. But before I go on, I need to know what you already know. Do you know how to evaluate an electric field at a point due to continuous charge distributions such as line of uniform charge and ring of uniform charge?

 

[K3nt70]

Well yes, but we're given specific equations for those. For example, E=2k lambda/r for an infinite line of continuous charge where r is a distance from that line. Continuous charge of a sphere (and you're looking for a point withing the sphere) is E = kQr/a^3 where a is the radius of the entire sphere and r is the distance from the center of the sphere to the point inside the sphere that you are looking for. This is pretty much the first time I've had to integrate on my own to get the formula for a question.

 

[Defennder]

I have no idea if you are required to learn how to integrate to find the electric field due to a uniform charge distribution, or simply know how to use derived formulae to calculate it. If it's the latter, then you should provide those formulae you think could help solve the problem, just post them along with an explanation of the variables in the formula and the geometric configuration the formula is valid for. The formula you posted for the infinite line of continuous charge can't be used here because the line here is finite and it is evident that neither the x or y components of the E-field is 0.

If it's the former, then I suggest you start by working out what $dE_y$ is first. r(x) is the distance between the origin and a point along the line of charge as it sweeps along the length of the line. Using pythagoras theorem, you can see that the distance is $$\sqrt{0.126^2 + x^2}$$.

The expression for $$dE_y$$ I have already given you, along with an explanation with why it is so. Now, what you need to do is find out what is $$\cos \theta$$ in terms of x and substitute that into the expression for $$dE_y$$. Once you're done, you should have an expression in terms of x and constants only. Integrate that from 0 to 44.2 cm and you have the y-component of the E-field. $$dE_x$$ can be found in a similar manner. Just think of how you can express $dE_x$ in terms of dE and $\theta$.

 

[K3nt70]

we don't need to derive the formula, that's just what i thought i had to do. I found this one:

$$\frac{kQ}{a(l + a)}$$

this was for a charged rod of finite length where the point is horizonal to the rod.
l is the length of the rod, a it the distance between the end of the rod and the point, Q is the charge and k is the constant.

 

[Defennder]

What do you mean by "horizontal to the rod"? Meaning to say it lies on the perpendicular bisector of the rod? Secondly in the question you are told to find the x and y-component of the E-field. But the formula you quoted only has one expression, which I suppose is for the magnitude of the resulting E-field or for either E_x or E_y, since the other would have been canceled out due to symmetry. Note that in this problem, you cannot make use of symmetry and must find both E_y and E_x.

 

[K3nt70]

The rod in the problem in the book is horizontal. the point is directly to the left of the rod.
Im fine with finding the magnitude of the magnetic field because the x and y components are easy to calculate with simple trig.

 

[Defennder]

By directly to the left end of the rod, do you mean to say that the rod and the point lie on the same line? If so, then clearly the formula you quoted isn't applicable, since again the geometric configuration of the rod and the point is very different in each case. More importantly, you can't simply extract the x and y components of the E-field from the magnitude since the x and y components are independent of each other.

My suggestion would be to start off with the expression for dE_y. But if this is not what you're supposed to do to solve the problem with what you've posted here, then I really can't help you out here.

 

[K3nt70]

Ok. You said that dEy was Cos$$\theta$$dE, right? and that

$$dE = \frac{\lambda dx}{4 \pi \varepsilon r(x)^2}$$

Meaning i have to perform the integral of:

$$dE = Cos\theta \frac{\lambda dx}{4 \pi \varepsilon r(x)^2}$$

I the theta i will be using is shown in the following diagram:

http://img174.imageshack.us/img174/8461/23991372qh3.png

 

[Defennder]

Ok. You said that dEy was Cos$$\theta$$dE, right?

That expression was for a different angle, more specifically it was for the other angle in the triangle you drew. Now since you're using that angle in the picture, use $$dE_y = \cos(\frac{\pi}{2} - \theta)$$ instead.

Meaning i have to perform the integral of:

$$dE = Cos\theta \frac{\lambda dx}{4 \pi \varepsilon r(x)^2}$$

Replace $\cos \theta$ with the expression for the angle given above and it's fine. Now find a way to express the trigo term cos(pi - theta) in terms of x. Use the diagram you drew. Once this is done, you have the integrand expressed in terms of x alone. Integrate that for the length of the rod and you have E_y

 

[K3nt70]

Wouldnt it make sense to use $$Sin \theta$$ ? Anyway these are the expressions i got for sin and cos:

$$Cos \theta = \frac{x}{\sqrt{x^2 + y^2}}$$
$$Sin \theta = \frac{y}{\sqrt{x^2 + y^2}}$$

So.. i need to take the integral of:

$$dE_x = \frac{x}{\sqrt{x^2 + y^2}}* \frac{\lambda dx}{4 \pi \varepsilon r(x)^2}$$

Correct?

 

[Defennder]

$$\cos ({\frac{\pi}{2} - \theta}) = \sin \theta$$

Yes that expression looks all right for $$dE_x$$. Now you have to plug in r(x) and integrate over the range of x. Do the same for $$dE_y$$ and you're done.

 

[K3nt70]

and $$r(x) = \sqrt{x^2 + y^2}$$ right?

 

[Defennder]

Yeah it is.

 

[K3nt70]

I'm having trouble completing the integral. I get this far:

$$dE_x = \frac{\lambda x}{4 \pi \epsilon (x^2 + y^2)} dx$$

 

[Defennder]

That expression is incorrect. What is r(x)^2 * sqrt(x^2+y^2)?

 

[K3nt70]

$$(x^2 + y^2) * \sqrt{x^2 + y^2} = x^2 \sqrt{x^2 + y^2} + y^2 \sqrt{x^2 + y^2}$$

 

[Defennder]

Or you make it a lot easier by writing it as $$(x^2 + y^2)^{3/2}$$. To integrate you may need to do a trigo substitution.