Find the x component of the Electric Field

[themagiciant95]

Homework Statement

Find the general formula for the x component of the electric
field if the charge density p varies only with x throughout all
space.

Homework Equations

The Attempt at a Solution

I started using the poisson equation $$\bigtriangledown \bullet \bar{E} = \frac{p}{\varepsilon_{0}}$$
From the problem text, I know that p has only the x component and so also E has only the x component.
With these information, i tried to calculate the poisson equation, obtaining :

$$\frac{\partial E_{x}}{\partial x} =\frac{p(x)}{\varepsilon_{0}}$$

But, i don't know how to continue the calculations... Can you help me ?

 

[phyzguy]

Since the variation is only in x, the left hand side is $\frac{dE_x}{dx}$, not $\frac{\partial E_x}{\partial x}$. Now multiply through by dx and integrate both sides.

 

[themagiciant95]

Since the variation is only in x, the left hand side is $\frac{dE_x}{dx}$, not $\frac{\partial E_x}{\partial x}$. Now multiply through by dx and integrate both sides.

Do i have to integrate it as an definite or indefinite integral ? In the latter case, how can i manage the constant of integration ? Thanks

 

[phyzguy]

Well, since you aren't told the boundary conditions, you will have to make some assumptions. You could assume for example that Ex at -∞ is zero. Or you could include the value of Ex at some point in your calculation.

 

[themagiciant95]

For example, if i include the value of Ex in a point (it's sufficient ?), how can i use this value to calculate the constant of integration ? Thanks again

 

[phyzguy]

Why don't you show us the calculation with the constant of integration included? Then if it isn't clear how to deal with it we can make suggestions.

 

[themagiciant95]

Why don't you show us the calculation with the constant of integration included? Then if it isn't clear how to deal with it we can make suggestions.

$$\int dE_{x} =\frac{1}{\varepsilon _{0}}\int p(x)dx$$

$$E_{x}(x)=\frac{1}{\varepsilon _{0}}P(x) + c$$

As boundary i chose:

$$x=0 \rightarrow E_{x}=0$$
so
$$c=-\frac{1}{\epsilon _{0}}P(0)$$

is this correct ? Thanks

 

[themagiciant95]

up

 

[phyzguy]

It looks OK to me. Without knowing more about the boundary conditions, I think something like this is the best you can do. You could try asking your teacher for more clarification.

 

[themagiciant95]

Thanks so much for your help :)