Question

Given :

\(\large \color{green}{r = x \hat{a_x} + y \hat{a_y} + z \hat{a_z}}\) and \(\large \color{blue}{r = |r|}\), then find the Divergence of \(\color{red}{\textsf{Vector A}}\) which is given as : \(\large \color{orange}{r^n r}\)..

Answer 1

\[A=r^n*r \\ \nabla A=\nabla r^n*r=\nabla (x^2+y^2+z^2)^{n/2}*(x i +yj+zk)\]

Answer 2

ok with this?

Answer 3

now
use the basic definition
\[\nabla F=[i \frac{ \delta F_x }{ \delta x }+j \frac{ \delta F_y }{ \delta y }+k \frac{ \delta F_z }{ \delta z }]\]

Answer 4

@waterineyes

Answer 5

Yes, I got that..

But when you put, then don't this derivative becomes lengthy and typical??

Answer 6

\[\frac{d}{dx}(x (x^2 + y^2 + z^2)^\frac{n}{2}) + \frac{d}{dy}(y(x^2 + y^2 +z^2)^\frac{n}{2}) + \frac{d}{dz}(z(x^2 + y^2 + z^2)^\frac{n}{2})\]

Answer 7

Is that right??

Answer 8

yeah it will be very lenghty
\[=[i \ \frac{ \partial }{ \partial x }+j \frac{ \partial }{ \partial y }+k \frac{ \partial }{ \partial z }]*[(x^2+y^2+z^2)^{n/2}*(x i)+\\(x^2+y^2+z^2)^{n/2}*(yj)+(x^2+y^2+z^2)^{n/2}*(zk)]\]

Answer 9

yes right!!

Answer 10

Now product rule and all that??

Answer 11

yeah :)

Answer 12

Okay, I try.. :)

Answer 13

okay !

Answer 14

I am still doing mistake somewhere.. :(

Answer 15

Check it if you have got this or not:

\[3(x^2 + y^2 + z^2)^\frac{n}{2} + n(x^2 + y^2 + z^2)^{\frac{n}{2} - 1}[x^2 + y^2 + z^2]\]

Answer 16

And this is giving me:

\[\color{red}{r^n(n+3)}\]

Answer 17

yeah i'm getting it
\[=(n/2)*(x^2+y^2+z^2)^{n/2-1}(2x^2)+(x^2+y^2+z^2)^{n/2} \\+(n/2)*(x^2+y^2+z^2)^{n/2-1}*(2y^2)+(x^2+y^2+z^2)^{n/2}+ \\(n/2)*(x^2+y^2+z^2)^{n/2-1}*(2z^2)+(x^2+y^2+z^2)^{n/2} \\=n(x^2+y^2+z^2)^{n/2-1}(x^2+y^2+z^2)+3(x^2+y^2+z^2)^{n/2}\\=(n+3)*(x^2+y^2+z^2)^{n/2}\]

Answer 18

Is their anything I am still missing??

Answer 19

yeah i'm also getting the same

Answer 20

But the answer is not that...

Answer 21

Four options given are:

\[a) \; \; (n+1) \cdot r^{{n-1}}\]
\[b) \; \; 3n \cdot r^{n-1}\]
\[c) \; \; (3n + 1) \cdot r^n\]
\[d) \; \; (nr + 3) \cdot r^{\frac{n}{2}}\]

And \(\color{green}{(c)}\) is supposed to be the answer...

Answer 22

Found something??:(

Answer 23

but i dont see any mistake here

Answer 24

Time to seek opinions from others masterminds too.. :)

Answer 25

@amistre64

Answer 26

yeah @SithsAndGiggles

Answer 27

@ganeshie8 your one look will be really helpful and needed here.. :)

Answer 28

hey there is really no mistake here

Answer 29

@waterineyes

Answer 30

I am here only..

Answer 31

pfft, im no master mind at this :) looks greek to me at the moment

Answer 32

@ganeshie8 on Facebook, I took help from @experimentX and he got option d) and I don't know how he got that..

But, his answer is well in the options provided and ours (me and Siddharth) is like God knows from where we have got this.. :P

Answer 33

It is okay, @amistre64 I am happy you replied.. :)

Answer 34

Well, I want to tell that It is a question from Schaum's Outlines of Electromagnetics, 2nd Edition..

Chapter 4 : Divergence And Divergence Theorem

Question No. 4.5 in Objective Type Questions, Also Page No is 4.14

Answer 35

Note : e-book, though 2nd edition, will not contain this question, I am talking about Hard Copy of that book and not Soft copy..:)

Answer 36

yeah i can have this book in our clg library i'll take a look at that :)

Answer 37

If you have access to Hard Copy of this book, then you must see that questions, May be answer choices are given wrong.. :) (Hopefully)

Answer 38

lol, yeah i also hope so :)

Answer 39

Okay @sidsiddhartha you truly helped a lot.. Thanks.. :)

Answer 40

np :)

Answer 41

and happy independence day :)

Answer 42

Yeah, Happy Independence Day to you @sidsiddhartha and @ganeshie8 .. :)

Answer 43

yeah @ganeshie8 too :)

Answer 44

Now, you can help other users here, and @sidsiddhartha do look for this question in that book.. :)

Answer 45

yeah i will definitely :)

Answer 46

Okay, that's really nice of you.. :)

Answer 47

:)

Answer 48

I want to bring it into notice that in some other book, the answer is : \((n+3)r^n\), so we are correct in finding that.. :)