Question

The electric potential in some region is expressed by \(\large \sf V=6x-8xy^2-8y+6yz-4z^2\) V.

The magnitude of force acting on a charge of 2C situated at the origin will be ?

Answer 1

Answer is 20 N.

Answer 2

I know i will have to find the field first but how to do that ?

Answer 3

@ash2326

Answer 4

E is gradient of potential
\[E=-\frac {\delta V}{\delta x}(i)-\frac {\delta V}{\delta y}(j)-\frac {\delta V}{\delta z}(k)\]

i, j and K are unit vectors in x, y and z direction.
Can you find now?

Answer 5

\[\frac{\delta V}{\delta x}= \text{partial differentiation of V w.r.t x}\]

Answer 6

ok..but what about the terms which contain both x and y...like -8XY\(^2\)

Answer 7

Sorry for my noobness....a bio base student

Answer 8

when you partially differentiate w.r.t x- all other variables are treated as constant
Say 8xy^2- partial differentiation w.r.t x--------------> 8y^2
partial differntiation w.r.t y-------------------------->18x*(2y)=16xy

Do you follow good sir?

Answer 9

ignore the 18, it's 8

Answer 10

Oh i got it.....So i will differentiate to get an equation for E, then multiply it with charge to get force...ryt ??

Answer 11

Once you find E from potential, you will get E in terms of x, y and z.

The question has stated another important fact- charge placed at origin.
So you need to find Electric field at origin- plugin x=0, y=0 and z=0
Once you have done that, you will have E at origin. Then you multiply with charge to find the force on it.

Answer 12

yes...yes...i follow you @ash2326 !

Thanx a ton !

Answer 13

Welcome :)

Answer 14

one min....can u tell me the final equation for E ?

Answer 15

I mean the result after differentiation.

Answer 16

What did you get? I will check it

Answer 17

So there will be three equations ryt ?

Answer 18

6-8y^2-8y+6yz-4z^2

Answer 19

6x-16xy-8+6z-4z^2

Answer 20

E= f(x,y,z) i+g(x,y,z)j+u(x,y,z)k

where f, g and u are functions of x, y and

Only one equation

Answer 21

ok can u show me an example..like differentiate some other random equation.

Answer 22

I will do for x
\[\large \sf V=6x-8xy^2-8y+6yz-4z^2\]

\[E=-\frac{\delta(6x-8xy^2-8y+6yz-4z^2)}{\delta x}+......\]
\[E=-(6-8y^2-0+0-0)i.....................\]
Do you follow?
you need to put the y direction and z direction field.

Answer 23

oh..so you differentiated it wrt x. I'll have to differentiate it wrt y and z. And finally put x,y,z as 0 since they are at origin.

Answer 24

yes, then you will get something of the form
\[E=-6i+aj+bk\]
for force, you will multiply by 2 Coloumbs

Answer 25

Another thing which is highlighted by the answer. It says 20 N, no direction!!!!
So you have to find the magnitude of the force. I believe you can do that

Answer 26

Yes i am getting
\(-6i+8j+0k\)
-->|E| = 10

So force=2*10=20N

Thanx for taking the pain @ash2326 . It's well said that doctors become the worse patient :D