How Do You Calculate Electrostatic Potential in a Decaying Electric Field?

[Alex_Neof]

Homework Statement

What is the electrostatic potential for the electric field in the region x ≥ 0 where:

E = (E0)*e^(kxi) V/m, with E0 a constant?

(The potential at x → infinity is defined to be zero).

Homework Equations

b
v = v_a - v_b = - ∫ E . dl
a

The Attempt at a Solution

0
v = - ∫ (E0)*e^(kx) dx = (E0)/k.
infinity

Is this correct?

Kind regards.

 

[BvU]

Helo Alex,

You can check for yourself if this is correct: how do you derive an electric field from a potential ? Does that work OK for your answer ?

Oh, and did I miss a minus sign somewhere ?

 

[Alex_Neof]

Thank you BvU !

 

[BvU]

Does that mean you found a potential $E_o\over k$ V gives an electric field $\vec E = E_0 \; e^{kx} \; {\bf \hat\imath} \$ V/m ?

 

[BvU]

Does that mean you found a potential $E_o\over k$ V gives an electric field $\vec E = E_0 \; e^{kx} \; {\bf \hat\imath} \$ V/m ?

 

[Alex_Neof]

So I was given the electric field:

$\overrightarrow{E} = {E_0} e^{-kx}\hat i.$

Which is completely in the x-direction.

Now using:

$V = {V_a} - {V_b} = - \int_a^b \overrightarrow{E}.\overrightarrow{dl},$

with the above electric field we have:

$V = - \int_\infty^x {E_0} e^{-kx}dx = \frac{E_0} {k} e^{-kx}.$

Then using what BvU suggested:

Helo Alex,

You can check for yourself if this is correct: how do you derive an electric field from a potential ? Does that work OK for your answer ?

Oh, and did I miss a minus sign somewhere ?

I remembered from my notes that the electric field is related to the potential as follows:

$\overrightarrow{E} = - \nabla V.$

So to test if the potential I found is correct, I just plug it in and find:

$- (\frac{\partial V} {\partial x})=-(-{E_0} e^{-kx}\hat i)$

Which gives us back our original Electric Field:

$\overrightarrow{E} = {E_0} e^{-kx}\hat i.$

Thanks again BvU, much appreciated!

 

[BvU]

well done !

 

[Alex_Neof]

Initially my limits for integration were wrong, I had them from $\infty$ to 0.