Testing paint skills( physics question)

Hi , i made this question yesterday, and it follows :

Consider an infinitely large wire (blue) having charge per unit length $+ \lambda$, and a finite length wire (red) of length $l$ and mass $m$ having the same charge per unit length. Its lower end rests at a height of $h$ units from the blue wire,as shown in the figure.

Image text

Your task is simple. Find the frequency of oscillation of red wire on being disturbed by a small distance vertically

#Physics #ElectricityAndMagnetism #PhysicsProblem

Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
`[example link](https://brilliant.org)`	example link
`> This is a quote`	This is a quote
# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"	# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

Hi Jatin!

Here's my solution:

At equilibrium,

$\displaystyle mg=\frac{\lambda^2}{2\pi \epsilon_0}\ln\left(\frac{h+l}{h}\right)$

Lets displace the red wire vertically upwards by $y$ .

The net force acting on the red wire is:

$\displaystyle F_{net}=\frac{\lambda^2}{2\pi\epsilon_0}\ln\left(1+\frac{l}{h+y}\right)-mg$

Using the approximation,

$\displaystyle \ln\left(1+\frac{l}{h+y}\right)\approx \ln\left(\frac{h+l}{h}\right)-\frac{ly}{h(l+h)}$

$\displaystyle F_{net}=-\frac{\lambda^2ly}{2\pi h(l+h)\epsilon_0} \Rightarrow \ddot{y}=-\frac{\lambda^2l}{2m\pi h(l+h)\epsilon_0}y$

The above equation is for SHM and the frequency can be easily deduced.

Is this correct?

I have a small question, how do you take such approximations? I had to use Wolfram Alpha, can you provide some help on this? Many thanks! :)

Pranav Arora - 7 years, 6 months ago

It was correct!, but you missed $m$ in the denominator, i do these approximations as:

$d(\ln (1 +\frac{l}{h})) = \frac{1}{1 + \frac{l}{h}} \times \frac{- l}{h^2} dh$ , where $dh = y$ , $d$ representing small change.

jatin yadav - 7 years, 6 months ago

Thanks Jatin! Yes, I missed the m, sorry about that, I will edit that. :)

I was wondering if you don't mind, could you please repost the same physics problem about force on mirror you posted before, I am very interested to know about its solution. Thanks! :)

Pranav Arora - 7 years, 6 months ago

@Pranav Arora – It was something where we had to apply gauss law in mirror problem, but since, i was a beginner that time in making problems i later noted that it had some problem, So, i had to delete it to avoid confusion, but still i can at least e-mail it at your id, which you might tell.

jatin yadav - 7 years, 6 months ago

@Jatin Yadav – Ah, I don't like to post my address in public but I remember you posted your e-mail id in one of the discussions. I remember your e-mail address so I will send you a mail. :)

Pranav Arora - 7 years, 6 months ago

You should also have shown how force of interaction is $\frac{\lambda^2}{2 \pi \epsilon_{0}} \ln\bigg(\frac{h + l}{h}\bigg)$ , I do it here,

Consider a small element of length $dx$ having charge $d q = \lambda dx$ at a distance $x$ from blue wire, we know that

$E_{x} = \frac{\lambda}{2 \pi \epsilon_{0} x}$

$F = \displaystyle \int_{h}^{h+l} E_{x} \lambda dx = \int_{h}^{h+l} \frac{\lambda}{2 \pi \epsilon_{0} x} \lambda dx$

= $\frac{\lambda^2}{2 \pi \epsilon_{0}} \ln\bigg(\frac{h + l}{h}\bigg)$

jatin yadav - 7 years, 6 months ago

How did you approximate it ??

Priyanka Banerjee - 7 years, 6 months ago

lambda/root(8pie^2epsilon zero*(h+l))...I think this is the answer..if solution is needed I will provide one later..a lil' bit busy!

Susovan Maity - 7 years, 6 months ago

No, this is not the answer.

jatin yadav - 7 years, 6 months ago

may have done something wrong during the approximation..sorry!

Susovan Maity - 7 years, 6 months ago

Challenge

Solve this within a minute:

We remove the blue wire and and take the red wire to a place where, electric field varies as $E =E_{0} e^{-x^2}$ It stays in equilibrium at some height $H$ say,and i repeat the same experiment, Task remains same , find the frequency of oscillations.

jatin yadav - 7 years, 6 months ago

S I unit of force

Vinay Chandra - 7 years, 1 month ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$