How to Find Electric Potential Difference Using a Given Electric Field?

[kacete]

Homework Statement

Given the Electric Field $$\vec{E}=40xy \vec{u_{x}} + 20x^{2} \vec{u_{y}} + 2 \vec{u_{z}} V/m$$ find the $$V_{PQ}$$, where $$P(1,-1,0)$$ and $$Q(2,1,3)$$.

Homework Equations

I know that:
$$V_{PQ}=-\int^{P}_{Q}\vec{E}d\vec{L}$$
I'm just confused, how to use it besides a straight line following a single direction $$\vec{u_{x}}$$ for example.

The Attempt at a Solution

I tried converting to spherical coordinates resulting in an equal confusion.
Solution (at the end of the book): $$106V$$.

 

[darkpsi]

You could think of it as a vector connecting the two points P and Q. Thus the "straight line" would be equal to the distance between the points in the direction pointing from P to Q.

 

[lubuntu]

Notice you can drop the z component from the calculation because there is no change in it from Q to P.

 

[kacete]

Notice you can drop the z component from the calculation because there is no change in it from Q to P.

I realized I made a typo, P is in fact (1,-1,0).

I thought of something like that, but if I integrate everything the result is a volume and not a line. In which direction should the integration be?

 

[darkpsi]

The direction would be composed of three different vectors. (Namely the vectors corresponding to the changes in the x, y, and z directions)

 

[kacete]

The direction would be composed of three different vectors. (Namely the vectors corresponding to the changes in the x, y, and z directions)

How about the beginning and end of the integral? Will it be $$0$$ and $$\sqrt{14}$$ (the module of the vector $$\vec{PQ}$$)?

 

[darkpsi]

Not quite. It would be easier if you changed the variables into just one variable and then you could integrate from, say, P_x to Q_x. I found the correct answer that way

 

[kacete]

Not quite. It would be easier if you changed the variables into just one variable and then you could integrate from, say, P_x to Q_x. I found the correct answer that way

How do you change the variables into just one?

 

[darkpsi]

Well you have to find how y and z vary with, say, x. If your electric field was just
E= xx+yy+zz then you could think of it as a triple integral and integrate over the three vectors that combine to create the vector u stated in your problem. But since your y-dependents vary with x, you have to find some way to relate the variables because your can't integrate a changing x as ∫xdy

 

[kacete]

Do you mean finding the vector component of the electric field in the direction of ux? For example.

 

[darkpsi]

Well it tells you the component in the x direction, it's 40xy. Maybe your confusing u_x as having an actual magnitude? The magnitude is the 40xy. u_x is simply saying the electric field (40xy) in the x direction, because the electric field is the same direction as vector u. To be accurate they really shouldn't even have a vector symbol above them but rather a hat because they are unit vectors. If you draw a simple grid and plot the two point (1,-1) and (2,1) you can see how y depends on x. Same goes for z. And then when the y's and the z's are in terms of x, you take the derivatives to get the dy and dz. Then you should have everything in terms of x's and dx's. Combine like terms and integrate.

 

[kacete]

I believe I understand now. Thank you very much for your help, I will try that.

 

[gabbagabbahey]

Well you have to find how y and z vary with, say, x. If your electric field was just
E= xx+yy+zz then you could think of it as a triple integral and integrate over the three vectors that combine to create the vector u stated in your problem. But since your y-dependents vary with x, you have to find some way to relate the variables because your can't integrate a changing x as ∫xdy

The electrostatic potential is path independent, and so the integration can be done over any path from $\textbf{P}$ to $\textbf{Q}$; not just the straight line path that you seem to imply is necessary.

Personally, I would break the path into three sections...one where only 'x' changes (so that $d\mathbf{\ell}=dx\mathbf{u}_x$), another where only 'y' changes (so that $d\mathbf{\ell}=dy\mathbf{u}_y$) and then a third where only 'z' changes (so that $d\mathbf{\ell}=dz\mathbf{u}_z$)

Of course, using the straight line from $\textbf{P}$ to $\textbf{Q}$ works as well, but it requires that you first parameterize that line.

 

[kacete]

Like this?

$$V_{PQ}=-\int^{P}_{Q}\vec{E}d\vec{L}=-(\int^{2}_{1}E\vec{u_{x}}dL\vec{u_{x}} + \int^{1}_{-1}E\vec{u_{y}}dL\vec{u_{y}} + \int^{3}_{0}E\vec{u_{y}}dL\vec{u_{y}})$$

 

[gabbagabbahey]

Like this?

$$V_{PQ}=-\int^{P}_{Q}\vec{E}d\vec{L}=-(\int^{2}_{1}E\vec{u_{x}}dL\vec{u_{x}} + \int^{1}_{-1}E\vec{u_{y}}dL\vec{u_{y}} + \int^{3}_{0}E\vec{u_{y}}dL\vec{u_{y}})$$

Your limits of integration are backwards,

$$V_{\textbf{PQ}}=-\int_{\textbf{P}}^{\textbf{Q}}\textbf{E}\cdot d\mathbf{\ell}=-\int_1^2 \textbf{E}\cdot dx\textbf{u}_x -\int_{-1}^1 \textbf{E}\cdot dy\textbf{u}_y-\int_0^3 \textbf{E}\cdot dz\textbf{u}_z$$

I suggest you draw a quick sketch of this path (remember you are starting at $\textbf{P}$ and finishing at $\textbf{Q}$)...what are the values of $y$ and $z$ along the first section of the path?...what are the values of $x$ and $z$ along the second section of the path?...what are the values of $x$ and [itex]y[/itrex] along the final section of the path?

 

[darkpsi]

The electrostatic potential is path independent, and so the integration can be done over any path from $\textbf{P}$ to $\textbf{Q}$; not just the straight line path that you seem to imply is necessary.

Personally, I would break the path into three sections...one where only 'x' changes (so that $d\mathbf{\ell}=dx\mathbf{u}_x$), another where only 'y' changes (so that $d\mathbf{\ell}=dy\mathbf{u}_y$) and then a third where only 'z' changes (so that $d\mathbf{\ell}=dz\mathbf{u}_z$)

Of course, using the straight line from $\textbf{P}$ to $\textbf{Q}$ works as well, but it requires that you first parameterize that line.

Yes but then when you dot E with dL you'll get
V_PQ = ∫∫∫ 40xy dx + 20x² dy + 2 dz
Am I missing something or does changing of variables need to take place?

 

[gabbagabbahey]

Yes but then when you dot E with dL you'll get
V_PQ = ∫∫∫ 40xy dx + 20x² dy + 2 dz
Am I missing something or does changing of variables need to take place?

You are missing something...draw out the path i suggested...along the first section, $y$ and $z$ will have a definite value...along the second section, $x$ and $z$ will have a definite value and along the last section, $x$ and $y$ will have a definite value.


Also, a path integral is quite different from a triple integral (if you are studying from Griffiths, I suggest you take a look at section 1.3.1!).

 

[kacete]

Your limits of integration are backwards,

$$V_{\textbf{PQ}}=-\int_{\textbf{P}}^{\textbf{Q}}\textbf{E}\cdot d\mathbf{\ell}=-\int_1^2 \textbf{E}\cdot dx\textbf{u}_x -\int_{-1}^1 \textbf{E}\cdot dy\textbf{u}_y-\int_0^3 \textbf{E}\cdot dz\textbf{u}_z$$

I suggest you draw a quick sketch of this path (remember you are starting at $\textbf{P}$ and finishing at $\textbf{Q}$)...what are the values of $y$ and $z$ along the first section of the path?...what are the values of $x$ and $z$ along the second section of the path?...what are the values of $x$ and [itex]y[/itrex] along the final section of the path?

Right! I understand it now. Yes, it's easier than to parametrize the line. I didn't think of that, but it makes sense. Thank you! I will try it.

 

[darkpsi]

You are missing something...draw out the path i suggested...along the first section, $y$ and $z$ will have a definite value...along the second section, $x$ and $z$ will have a definite value and along the last section, $x$ and $y$ will have a definite value.


Also, a path integral is quite different from a triple integral (if you are studying from Griffiths, I suggest you take a look at section 1.3.1!).

Well just for future reference, when couldn't I change everything to one variable to solve?

 

[gabbagabbahey]

Well just for future reference, when couldn't I change everything to one variable to solve?

Any curve can be written in terms of a single parameter (variable), so it is always possible to parametrize the straight line joining the two endpoints and do a single integration...However, it is often easier to choose a different path and avoid the parameterization process.

 

[darkpsi]

Any curve can be written in terms of a single parameter (variable), so it is always possible to parametrize the straight line joining the two endpoints and do a single integration...However, it is often easier to choose a different path and avoid the parameterization process.

Oh I see. I'm actually learning this right now, so that's nice to know. I tried it the way you suggested and it does seem less tedious.

 

[kacete]

I managed to solve to the right solution, except that I get a negative value: -106 instead of 106. What did I do wrong? Or is this expected?

 

[darkpsi]

I managed to solve to the right solution, except that I get a negative value: -106 instead of 106. What did I do wrong? Or is this expected?

either you went from Q to P instead of the other way, or you forgot to take the negative of the integral

 

[kacete]

Probably the second. Are you supposed to take the negative of the integral out?

 

[darkpsi]

Probably the second. Are you supposed to take the negative of the integral out?

since it's a constant of integration, it wouldn't matter either way.