Exploring the Inconsistencies of Work and Energy on an Infinitely Thick Sheet

[A13235378]

Homework Statement

Consider an infinitely thick sheet A positively charged with load density P. Also, consider that the coordinate system is located exactly in the center, with the x axis parallel to the thickness. Given a negative charge that is at X = A / 2 with a velocity V. What will be your velocity when reaching the origin?

Relevant Equations

E= Px/e (field inside the blade). , where e=permissiveness electric
Theorem of work and energy

My doubt is in the mathematics part. As the force will be contrary to the direction of the field due to the negative charge, the particle will be attracted to the origin, so I assumed that the speed will increase (this is also found in the exercise response). I then applied an integral for the job formula, from the limits (x = A / 2) to (x = 0):

$$W=\frac{Pqx^2}{2e}-->W=0-\frac{PqA^2}{8e}$$

This result shows that the work is negative, that is, by the energy work theorem, the final speed will be less than the initial one, being inconsistent.

Where I missing?

 

[BvU]

Homework Statement:: Consider an infinitely thick sheet A positively charged with load density P. Also, consider that the coordinate system is located exactly in the center, with the x-axis parallel to the thickness. Given a negative charge that is at X = A / 2 with a velocity V. What will be your velocity when reaching the origin?
Relevant Equations:: E= Px/e (field inside the blade). , where e=permissiveness electric
Theorem of work and energy

Where I missing

Is this the verbatim rendering of the problem statement of this exercise?
Is it about a sheet or a blade ? Infinitely thick ? Or an infinite thick sheet ?
Can you sketch the situation somewhat ?
Thickness is a scalar -- an axis is a vector, so where is the axis ?
Why would there be a field ?
And: why would you reach the origin ?

$\$

 

[A13235378]

Is this the verbatim rendering of the problem statement of this exercise?
Is it about a sheet or a blade ? Infinitely thick ? Or an infinite thick sheet ?
Can you sketch the situation somewhat ?
Thickness is a scalar -- an axis is a vector, so where is the axis ?
Why would there be a field ?
And: why would you reach the origin ?

Sorry for my concordance errors. English is not my native language, so I need to use a translator. I hope the drawing improves what I mean.

Since the plate is positively charged, and we have a negative charge, the particle will be attracted. The field inside has already been given, varying with the distance and having direction on the x axis. The question asks for speed at the origin. My doubt is that the final speed, according to the answer, will be greater than the initial speed. But I ended up finding a slower speed with my calculation. Where am I going wrong?

 

[BvU]

English is not my native language

OK, we will get there.
I suppose the <...> means that he sheet goes on to infinity.
And the field inside the sheet is indeed $\rho x/\varepsilon$
So there is a force pulling the negative charge towards $x = 0$. (Just like with a spring).

But I ended up finding a slower speed with my calculation. Where am I going wrong?

That is hard to say: you don't post your work !
[edit] let me look at post #1 again...

$\$

 

[TSny]

$$W=0-\frac{PqA^2}{8e}$$

This result shows that the work is negative

Does your expression for $W$ come out to be negative if you take into account that $q$ is a negative number?

 

[BvU]

And don't forget the initial $v\ \$ !

 

[haruspex]

Homework Statement:: Consider an infinitely thick sheet A

I suggest you mean an infinite sheet of thickness A.