# Electric Field direction?

1. Nov 11, 2007

### PFStudent

1. The problem statement, all variables and given/known data

7. In Fig. 28-35, an electron accelerated from rest through potential difference $${V}_{1} = 1.00{{.}}kV$$ enters the gap between two parallel plates having separation $$d = 20.0{{.}}mm$$ and potential difference $${V}_{2} = 100{{.}}V$$. The lower plate is at the lower potential. Neglect fringing and assume that the electron’s velocity vector is perpendicular to the electric field vector between the plates. In unit-vector notation, what uniform magnetic field allows the electron to travel in a straight line in the gap.

2. Relevant equations

$${\vec{E}}_{p1} = {\frac{\vec{F}_{21}}{q_{2}}}$$

$${\Delta{V}_{p}} = {-}{\int_{r_{0}}^{r_{1}}}{\vec{E}_{p1}(r)}{\cdot}{d{\vec{r}}}$$

$${\vec{F}_{B}} = q{\vec{v}}{\times}{\vec{B}}$$

3. The attempt at a solution

Ok, this problem does not seem that hard, however the wording isn’t really clear. So let me see if I understand this problem, you have an electron being accelerated by $${V}_{1}$$ in to an electric field produced by two plates. Where the bottom plate has potential $${V}_{2}$$.

Ok, so my question is if the bottom plate has potential $${V}_{2}$$, does the top plate have a potential?

Also, what about the electric field between the two plates how can you tell in what direction the electric field points in?

Any help is appreciated.

Thanks,

-PFStudent

Last edited by a moderator: Apr 23, 2017
2. Nov 12, 2007

### PFStudent

Hey,

So, I was looking at this problem again and I am still not all that sure about how you can tell which way the electric field points? How is that you determine that in this problem?

Any help is appreciated.

Thanks,

-PFStudent

3. Nov 12, 2007

### Staff: Mentor

The electric field points from higher to lower potential, which can be seen from this equation of yours (that's the meaning of the minus sign):

4. Nov 26, 2007

### PFStudent

Hey,

I was looking at this problem again and had a quick question. When finding the potential difference,

$${\Delta{V}_{p}} = {-}{\int_{r_{0}}^{r_{1}}}{\vec{E}_{p1}(r)}{\cdot}{d{ \vec{r}}}$$

Do you always integrate from the lower potential to the higher potential? In other words when finding the Electric field would I integrate from,

$${r_{0}} = {-{\frac{d}{2}}}$$

to

$${r_{1}} = {+{\frac{d}{2}}}$$

So, that the integral would look like the following,

$${\Delta{V}_{2}} = {-}{\int_{{-{\frac{d}{2}}}}^{{{\frac{d}{2}}}}}{|{\vec{E}}_{}(r)|}{}{|d{{\vec{r}}}|}{cos\theta}$$

Where, $${\theta} = {180}^{o}$$, since $\vec{E}$ points downward (from the higher potential to the lower potential) and $d{\vec{r}}$ points upward from ${r_{0}}$ to ${r_{1}}$.

Is that right?

Thanks,

-PFStudent

Last edited: Nov 26, 2007
5. Nov 28, 2007

### Staff: Mentor

Note that you are always finding the potential of r_1 with respect to r_0. If you reverse the integration (integrating from r_1 to r_0) the sign changes:

$$\int_a^b Fdx = -\int_b^a Fdx$$

So it doesn't really matter which way you integrate. What does matter is the direction of $\vec{E}$ compared to $d\vec{r}$.

6. Nov 28, 2007

### PFStudent

Hey,

Forgot to say thanks for the above info. That information had helped me better understand how to solve these type of problems. Thanks.

Ah...yes I do remember that. Thanks for reminding me.

Ok, so then on the below integral,

Do I have the limits correct?

Ok, so was I correct in performing the integration the way I did, Where ${\theta} = {180}^{o}$, since $\vec{E}$ points downward (from the higher potential to the lower potential) and $d{\vec{r}}$ points upward from ${r_{0}}$ to ${r_{1}}$?

Also, since you mentioned that propery of integrals--I could alternatively intregrate the following way,

$${\Delta{V}_{2}} = {}{\int_{{{\frac{d}{2}}}}^{{{-}{\frac{d}{2}}}}}{|{\vec{E}}_{}(r)|}{}{|d{{\vec{r}}}|}{cos\theta}$$

Is the above integral correct also?

So, in the end was the below integral correct?

$${\Delta{V}_{2}} = {-}{\int_{{-{\frac{d}{2}}}}^{{{\frac{d}{2}}}}}{|{\vec{E}}_{}(r)|}{}{|d{{\vec{r}}}|}{cos\theta}$$

Thanks so much for the help,

-PFStudent

7. Nov 28, 2007

### Staff: Mentor

Yes. You are finding the potential difference of r_1 with respect to r_0.

Absolutely.

Yes indeed.

You bet.

8. Nov 29, 2007

### PFStudent

Hey,

Thanks for the help Doc Al!

Thanks,

-PFStudent

Last edited: Nov 29, 2007