# Energy in a volume

DODGEVIPER13

## Homework Statement

If V=2x^2+6y^2 V in free space, find the energy stored in a volume defined by -1<x<1,-1<y<y, and -1<z<1. (BTW the < are suppose to less than or equal to or greater than or equal to)

## The Attempt at a Solution

I am not really sure of the formula here but I assume it needed at triple integral as it gave me 3 sets of limits and a function. Furthermore it asked for a volume so it was the only thing I could think of I relize it is probably wrong but am I even close? If not what formula should I use?

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Mentor
A triple integral is good, but you have to find out what to integrate first.

I'm not sure how to interpret "V=2x^2+6y^2 V", but it looks like an electric potential, so we have an electric field in that volume. What is the energy density of an electric field?
Are you sure there are no units involved?

DODGEVIPER13
Oh sorry yah the V stands or volts so yes it is electric potential. (1/2)(epsilon)E^2 equals energy density of an electric field

Mentor
Good, now you can use that to solve the problem.

DODGEVIPER13
So what your saying is that E would equal electric potential here or should I use dV=-Es(dS)

DODGEVIPER13
Thus I would take the derivative of the voltage

Mentor
Thus I would take the derivative of the voltage
That's the idea, right.

DODGEVIPER13
Well what should I derive the function by x or y I'm guessing x

Mentor
Well what should I derive the function by x or y I'm guessing x

x,y, and z, actually. You want to determine the electric field, a vector quantity, by finding the gradient of the potential. Presumably you'll then use the magnitude of this vector in your energy density evaluation (triple integral over the volume).

DODGEVIPER13
Ah ok man thanks so partial derivative with respect to x y and z then integrate that three times with the limits given got it will do later on

DODGEVIPER13
Well the gradient is (4x,12y) if I integrate it straight I get 0 which can't be right . I don't know if it is mathematically legal for me to e the agnitude with the x and y arable still in it but ill try anyway see if that does the trick I gets 101.19288

Mentor
Remember that you have to square its magnitude:
Oh sorry yah the V stands or volts so yes it is electric potential. (1/2)(epsilon)E^2 equals energy density of an electric field

DODGEVIPER13
6e-9

DODGEVIPER13
I got the magnitude to be sqrt(4^2+12^2) which is sqrt(160)=E so then E^2=160 which when I integrate I get 6e-9

DODGEVIPER13
Where epsilon is 8.85e-12

Mentor
(4x,12y) depends on x and y, and so does the magnitude. In addition, you are missing the z-component here (it is zero, but you have to include it to get a vector with 3 components).

DODGEVIPER13
I get E=sqrt((4x)^2+(12y)^2)=sqrt(16x^2+144y^2) then E^2=16x^2+144y^2 after integrating I get (1/2)epsilon(1280/3)=2e-9

DODGEVIPER13
Am I ok not trying to rush you just wondering?

Mentor
You can check calculations with a computer, so what are you waiting for?
The answer has missing units, but the problem statement has the same problem.

DODGEVIPER13
Ok thanks man yah u are right wolfram said it was ok. I guess I said it wrong did I have the integral set up correctly so that when I performed the integration it would be correct?

Mentor
That looks good.

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DODGEVIPER13
Ok thanks man

DODGEVIPER13
oh hey I know this post is old but I caught a mistake I believe since dV=-E dot dS then the answer shourld be negative 2E-9 also what should the units be Joules?

Mentor
You square the electric field, its direction does not matter, and the energy density is always positive.

DODGEVIPER13
Ah yah your right didnt consider that it was squared.