88 Replies
@Gyro Gearloose
Note for OP
+solved @user1 @user2... to close the thread when your doubt is solved. Mention the users who helped you solve the doubt. This will be added to their stats.Direct Laplace equation
the what eqn
$$\nabla^{2}V = \frac{-\rho}{\epsilon}$$
In your case, since V doesn't depend on theta and phi, it simplifies
in which chap is this eqn in book i hve never seen this
It's Gauss' Law in differential form combined with E = -dV/dr
OOPS
Wait
Opt
never seen this i need to read up more on this stuff
Sorry, I missed the minus sign
It's fine, you can do without that
thnk god because partial differntiations just... scare me
You just do it in two steps basically
Here because there's no theta or phi, it's a simple derivative only
thts great though
The hell is all this
Laplace was in Thermodynamics
i suck at simple diffrential at times
Iirc
$$\frac{d^{2}V}{dr^{2}}=\frac{-\rho}{\epsilon}$$
Opt
exct my rxn
so we integrate both sides to get the result here?
No, no need
It's rho that is asked
So differentiate V twice and multiply by epsilon
Should give answer
aah easy
i will try now
Opt bhai samaanya aam simple bhasha me bata do kuch
Ok, I'll explain it in a way that's mathematically wrong, but simple to understand
E × A = q/e (I'll use e for epsilon)
if i am not diffrential dumb it should be 4a (epslon not)
No ye to gauss law hai
😭 final ans bolri hu
E × A × e = q
Divide by volume
E × e/l = rho
But E = -V/l
-V/l × e/l = rho
Rho = -e(d²V/dr²)
Dimensions are correct
Woah
Crwazy
mujhe isse samjhne do
10 min lagega
How are supposed to figure this out in the exam again?
💀💀💀
good question
Any person who has studied vector calc will sue me for this bakwaas explanation
I fortunately haven't so yea lucky day
i didnt get the last step
Actual explanation is that Gauss' Law, which is integral as taught to you guys, can also be written in a differential form.
how it turned to diffrential
Dimensions ke anusaar 🤓
As I said, it's wrong.
Mai yahi likhne wala tha
Ahhhhh
Right
I can try to explain
The actual thing
If you guys really want
NVMMM I NEED SLEEP
KAL
woops
Or you remember this
shut up let me learn and u also learn
Preferably this
ye bataya to tha hume ek baar
Like just mentioned
Never thought question me use aaega
See if V depends on x,y,z use the ulta triangle
i saw the actual eqn from google and its scary
If it depends only on r, use the d²/dr² eqn
You saw the full ulta triangle in spherical coords
wait something like tht is possiblle
ys
Ohhhh right diffrnetial form of the gauss law ok
1/r(d²/dtheta²( rV)) kuch hoga na?
And 1/rsintheta (d²/dphi²(rsinthetaV))
I'm not sure that's right
I don't remember the Laplacian in spherical
so the actual guass law is something like this?
Yup
Well, both are Gauss' Law
i definetly need to learn derivatives more and then this version
It's just that if you integrated this one, you'd get the high school wala one
No.
You will go crazy
one we see is modified
Pls don't for the sake of mental health
ok okk
nhi karungii
Not really.
aise hi maut aayi hui hai
modified as in integrated to make it simpler
let me write this on my notes
Remember this applies if V is a function of only r and not any other variable of space like x,y,z,θ, or φ
If V is a function of just x, y, z, then you use this instead
$$\frac{\partial^{2}V}{\partial x^{2}}+\frac{\partial^{2}V}{\partial y^{2}}+\frac{\partial^{2}V}{\partial z^{2}}= \frac{-\rho}{\epsilon}$$
Opt
i hope i dont have to use ths ever cuz i have no idea how to solve this
You don't solve anything
then how does one use this eqn
It'll just give you rho as a function
ooohhh
Why are u scaring ppl bro just say gauss law 😭😭
Gauss law says the 'divergence' of electric field is rho/epsilon
there are 2 things defined for fields divergence and curl and those make up maxwell 4 laws
You don't need to know what they are to solve them tho, only if ur doing physics or math in the future you would use this
Ans is 4 ?
12
Ok I'll do this then
Mera to -4 aara 💀💀💀
Doesn't differential form of gauss law gives 4
E= -dV/dr
Divergen of E = rho/epsilon
Ohhh right
Bruhh online E= rho r /3epsilon kar dia wtf
Aayein
💀💀💀
Isn't that for uniform charge density
Merko toh 4 hi lag rha 12 just seems wrong
Do check by standard method tho
Phi(out)- phi(in) karke
But isme E ka kyu use
Hume seedhe seedhe
V diya hai
Double differentiate karke
-p/e se kyu n equate kar de
Why are we bringing in E here
Brother V ka first derivative E Hi toh hota hai
Uska partial derivative lete
We both r saying the same thing lol
dV/dr is E
But we are equating d²V/dr²
Isme E kuch bhi ho hume V to de hi rakha hai
neither so i think im good
Dude am so dumb
E=-dV/dr = -4ar
Now r is a vector and we have to find it's divergence not simple differentiation but partial
r = xi+yj+zk
E= -4ar = -4a(xi yj zk)
Now take the del/delx and stuff
Rho/epsilon = -12a
Yeah spherical coords are evil
Mb I made a mistake last night
WOAH WHATTTT
FR?
YE TO MUJHE NAHI PATA THA
My sincere apologies for being so wrong last night 🙇♂️
Look at the photo I sent above
Bro escaped from plain sight
us
khair agr apko smaj aa gya hai to doubt band kar dijiyega
yee goteem
thanks
+solved @Opt @hardcoreisdead
Post locked and archived successfully!
Archived by
<@964432960197632059> (964432960197632059)
Time
<t:1748784470:R>
Solved by
<@763645886500175892> (763645886500175892), <@741159941934415883> (741159941934415883)