Stuck on (proving) a linear charge problem

[FlyingButtress]

Homework Statement

Question:[/B]

____________________________________________________________________________________

And the answer in the solution manual:

Homework Equations

Trig sub? $$a^2/x^2 = tan(θ)$$
I'm thinking x>>a would affect this majorly and I'm not seeing how.

The Attempt at a Solution

I understand the concept behind this (I think), but part c is killing me right now. I have no idea how the inverse square root turns into the final equation. Maybe I should have posted this in Calculus, but I have no recollection of learning this in calc I/II, nor physics I/II so I didn't know where to post.

The $$a^2/x^2$$ looks eerily similar to a trig sub problem, but I don't see how that would work because there's no integral.

For $$E_x,$$ I was able to set the limit of a to 0 to get the point charge equation. The solution manual didn't do it this way but it worked for me to get the point charge equation.

BTW I already have a) and b), which were pretty straightforward.
Could someone point me in the right direction please? Thank you!

 

[tms]

Homework Equations

Trig sub? $$a^2/x^2 = tan(θ)$$

You should double-check that with the definition of the tangent.

3. The Attempt at a Solution

I understand the concept behind this (I think), but part c is killing me right now. I have no idea how the inverse square root turns into the final equation. Maybe I should have posted this in Calculus, but I have no recollection of learning this in calc I/II, nor physics I/II so I didn't know where to post.

The $$a^2/x^2$$ looks eerily similar to a trig sub problem, but I don't see how that would work because there's no integral.

For $$E_x,$$ I was able to set the limit of a to 0 to get the point charge equation. The solution manual didn't do it this way but it worked for me to get the point charge equation.

BTW I already have a) and b), which were pretty straightforward.
Could someone point me in the right direction please? Thank you!

Assuming you have the first two parts, you might want to take a look at the Taylor expansion of those answers.

 

[vanhees71]

How do you come to the conclusion that your equation is relevant for the problem? You should rather think about the question, how to calculate the force between two point charges and then how you can generalize this to many point charges and finally to a continuous charge distribution!

 

[FlyingButtress]

You should double-check that with the definition of the tangent.Assuming you have the first two parts, you might want to take a look at the Taylor expansion of those answers.

Ah, okay. The format change to (1+x)^(n) is making more sense now... I'm very rusty on infinite series so I'll have to take some time to review. From a brief look at my calc book it looks like it matches the binomial series. Will spend more time to see if it is right. Thanks!

How do you come to the conclusion that your equation is relevant for the problem? You should rather think about the question, how to calculate the force between two point charges and then how you can generalize this to many point charges and finally to a continuous charge distribution!

They just looked similar, like $$y^2/x^2$$
No other reason.
I actually did use that for part a, solving for $$E_x$$, but I've already solved parts a and b :) Sorry if I wasn't clear.

 

[HalfLostAndSearching]

It seems like you are struggling with the mathematical manipulation of the equation in part c. This can be a common issue when dealing with complex equations in physics. My suggestion would be to break down the equation and try to identify any patterns or relationships that may help you simplify it. Also, it may be helpful to review any relevant mathematical concepts that you may have learned in previous courses. Additionally, don't hesitate to reach out to your professor or classmates for assistance. Sometimes discussing the problem with others can help you see it in a different way. Keep persevering and don't get discouraged, as finding solutions to challenging problems is a crucial part of being a scientist. Good luck!