Force Calculation

[glad portal]

Yeah so, obviously the feild on the junction by either of the strips approaches infinity but how come the forces are finite? im confused, alot of questions im doint have a divergent integral but answers are finit

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[glad portal]

this is what ive done

[uneven heron]

I assume they meant force per unit length, as indicated by the units of each option (N/m).

The total force is infinite, of course.

So, the force from one whole half acting on a rod of unit thicnkness of the second half, I guess.

[glad portal]

even if its per unit length, at the junction the feild is supposed to be infinite

[uneven heron]

Hm...

[glad portal]

so if i sum forces on that unit length, it should give infinite

[uneven heron]

Let's see. I'll use (x, y) for the coordinates on the left strip and (u, v) on the second.
First, let's find the force acting from an element of the second strip on the element of the first.
dF = k dq1 dq2 e({u - x, v - y})/((u - x)^2 + (v - y)^2)^(3/2) = -kσ^2 dx dy du dv e({u - x, v - y})/((u - x)^2 + (v - y)^2)^(3/2)
We want dF/dy, as discussed above.
dF/dy = -kσ^2 dx du dv e({u - x, v - y})/((u - x)^2 + (v - y)^2)^(3/2)
Let's only focus on the horizontal component, since the vertical one certainly integrates to 0.
dF(hor)/dy = -kσ^2 dx du dv (u - x)/((u - x)^2 + (v - y)^2)^(3/2)
And then it's a triple integral: x from 0 to b, u from b to 2b, v from -∞ to ∞. That will take a while...

Also, yeah, I do agree that it seems to diverge.

Hm...

Although...

Yeah, diverges.

Even in case of dF/dy.

[glad portal]

yeah this is what ive done but in a simplified manner, diverges anyway

[glad portal]

exactly

[glad portal]

i dont know what exactly i've calculated here

but it seems to get me the answer

ill just go through and check what ive exactly done

[uneven heron]

This is what I'm getting. So, certainly diverges.

Odd... Then I'm not really sure what they're asking.

Maybe they want dF/(dxdy)?

But no, the units don't work out, I think.

Wait, hold up.

[uneven heron]

Now I'm not as sure whether this conclusion is correct.

It's just that I've tried another order of integration, and it does seem to converge now. Hm...

Huh. And now it works...

I'm going to be busy for a while, but I'll show what I got later.

[glad portal]

oki

[glad portal]

i reviewed my solution

i unconsciously calculated the force per unit length

but idk how mne's coming convergent

[glad portal]

yee

[uneven heron]

Here's what I'm getting now.

[uneven heron]

So, where's the problem here...

[glad portal]

i dont think we needed to complicate this simple question to this extent 😭

[uneven heron]

I think the problem is in the last step. Writing areasines in terms of logarithms would've probably been better.

I think the problem is due to the fact that v - y isn't of constant sign, so the way I wrote the antiderivative is wrong.

[uneven heron]

Well, I mean, that's my only idea of how to do it.

[glad portal]

this question probablyyy requires 2-3 minutes of manual solving and i doubt if its possible this way\

[uneven heron]

Really, 2-3 minutes? Hm...

[glad portal]

yea

the way i did iyt

[uneven heron]

I don't think I'd be able to do it that quickly.

[glad portal]

it

its a problem from an exam

where on an average we have 2-3 minutes per question

[uneven heron]

Woah...

Very odd.

[glad portal]

i have no problem with me getting the answer this way mathematically, i wouldve ticked the correct option and moved on

but intuitively i dont feel like the force is finite

ill have my teacher look over this and update you

[uneven heron]

I mean, I don't really care about the options. I'm just wondering where I went wrong in that last step during the first time.

Let me try writing it another way.

[glad portal]

electro stats are weird

the deeper i go

the more divergent inegrals i come across

each one of them has a weird ass "approximation"

[uneven heron]

Ah, I think I see what the problem might be with the order I picked first.
If you just naively paste the limits in the last step, you get this. But the red tem is 0 for v < 0, which causes problems. So, the limits here must be taken more carefully.

[uneven heron]

Well, this one isn't divergent, you just have to be very careful in how you take it.

In case of improper multiple integrals (that converge) one order of integration may be easier to deal with than others.

So, I guess I was just unlucky to pick a difficult one at first 😅

[glad portal]

so i was lucky in picking the right order?? 😭

[uneven heron]

Well, your approach isn't 100% transparent to me, but yeah, seems so.

[glad portal]

what i did was calculate feild at any distance "a" from the edge of one strip

with that feild, i took an elemental rod at distance "a" from the edge of the strip of width da and length L

charge on that rod is sigma. da. L

[uneven heron]

Yeah, I think I see. Your approach is similar to the order dxdvdu.

Which works a bit better, since dvdudx is quite a bit harder, as we just saw.

[glad portal]

dividing by L (for force per unit length) the L cancels

[glad portal]

yeahh

there was no logical or intuitive explanation for this

just mathematical ease

[uneven heron]

Well, as long as it's correct, why not?

[glad portal]

i dont feel right until i get the intuition correct 😭

[uneven heron]

Try writing some comments in your solution. Helps sometimes.

[glad portal]

noted

anyways thankyouuuu ill try changing the order for another question where im getting a divergent integral

maybe it'll help

[uneven heron]

You're welcome!

It's pretty interesting. I think I also encountered a similar situation when I learned physics, but this was a while ago.

[glad portal]

highers physics is wild

alr im closing the ticket

thankyouu againn

were you typing somthing

+close

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