In a 3-D space, the potential depends only on 'x' and is given by V(x,y,z)=-2*x^n. In other words, the potential is proportional to the nth power of the x coordinate.
In this case, find P(x,y,z), where 'P' stands for the volume charge density.
Click here for the solution
is the charge density -8*pi*epsilon*n*(x^n)/yz ?
ReplyDelete-n*x ki power(n-2)*epsilon
ReplyDelete-4 epsilon n^2 x^(n-2)
ReplyDelete-2n(n-1)x^(n-2)ε ?
ReplyDeleteThe ans. has to be -2n(n-1)є x^(n-2)
ReplyDeletewhat is the approach used ?
ReplyDeleteplease don't post the solutions/approach yet.
ReplyDelete@sambhav: could you point out which of these answers is correct?
to be precise, none
ReplyDeleteI think people didn't notice a minus already in the potential function
I too missed the minus sign in the potential function..:-(
ReplyDeletethe correct answer should be 2n(n-1)εx^(n-2)
What happened to the 'solutions on the way'?
ReplyDeleteSomeone please post the solution or just comment if Mayank's answer is correct..
yes, Anonymous, Mayank's answer is correct. I was waiting for more replies, I hope i have the right to do that.
ReplyDeletebtw, it would help us bloggers if you use some account to post comments, please don't post anonymously.
Solution will be posted soon.
Lets try a simple variation:
ReplyDeleteIn a 3D space, the potential depends on the distance 'r' from the origin as:
V(r)=a*r^2 + b*r +c (a quadratic variation).
Find P(r).
is the answer -6aε ??to the variation .
ReplyDeleteplz confirm ....
ReplyDeleteP(r)= -2ε(3ar+b)/r ?
ReplyDeletei don't remember the differentiation of a particular class of integrals (which is what i used to use to solve these problems).
ReplyDeleteso i can't post the correct answer. we'll wait for Sambhav.
if possible, try to find the field corresponding to this charge distribution and see if its correct.
-2ε(3ar+2b)/r
ReplyDeletevery well done Mayank, the 'variation' wasn't as easy!
ReplyDeleteI'll try to post both the Mathematical solution, as well as the one using Gauss' Law soon :-)
Rishabh, calc mistake, eh?
arre aap sab ne yeh answer position vector 'r' i terms mein kyon leave kar diya ..
ReplyDeletei resolved the position vector in three components and then solved to get -6aε........
firstly, 'r' here is not the position vector, it is the distance from origin.
ReplyDeletesecondly, I solved this question by two methods, and writing 'r' as sqrt(x^2 + y^2 + z^2) is the mathematical solution i was talking about.
okay i took it as position vector ..
ReplyDeletehad it been position vector then what would have been the answer ?
plz reply .......
that does not make sense to me: it would be smth like:
ReplyDeletescalar(potential) = vector (a vector expression in 'r' vector)
arre mera matlab yeh nahi hai ..
ReplyDeletei assumed 'r' to be position vector in the expression of electric field and replaced it with x i cap +y j cap+z k cap. and then choose a differential cube of dimensions dx * dy * dz then find flux thru it and apply gauss law to get -6a epsilon as answer........
where did i go wrong??????????
arey mere bhai, potential is a scalar, so ar^2 + br + c should also be a scalar, which cannot be if r is a vector
ReplyDeleteand moreover, you can't square a vector, can you?
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ReplyDelete@ rajiv
ReplyDeletemaine potential ke expression mein r ko vector nahi assume kiya . when u differentiate it u get 2ar+br which the expression of the electric field . in this expression i replaced r as x i cap + y j cap + z k cap (I CAN SURELY DO THIS AS ELECTRIC FIELD IS A VECTOR AND 'r' IS THE DISTANCE FROM THE ORIGIN i.e. IN VECTOR NOTATION I CAN WRITE IT AS MAGNITUDE OF POSITION VECTOR "r" ...................... )
and for ur kind information square of a vector means its dot product with itself .......
how can u forget such a simple concept ??????????????
No finger-pointing here please.
ReplyDelete@Abhishek: differentiation gives: 2ar+b. even here its not possible to consider 'r' a vector.
Lets solve this anyway:
Whats the charge distribution that produces a field:
p(x`)+q(y`)+r(z`) (a vector field).
x` means x cap.
WHY IT IS NOT possible to consider 'r'as a vector ?
ReplyDeleteBECAUSE B IS A SCALAR!
ReplyDeleteis this a constant field bhaiya?? or were u talking about:
ReplyDeletepx i + qy j + rz k??
how to solve this ;;;;;;;;;;;;;;;;;;
ReplyDeleteLets solve this anyway:
Whats the charge distribution that produces a field:
p(x`)+q(y`)+r(z`) (a vector field).
x` means x cap.
its a constant field. only infinite planes produce constant fields.
ReplyDeleteso that answer will be an infinite plane whose orientation is perpendicular to p(x`)+q(y`)+r(z`)