# Find focal length of electron for a parabolic motion

• Istiak
In summary, the conversation is about using the equation ##\int \vec F \cdot d\vec l = \frac{1}{2}mu^2## to find the value of ##l##. The person initially uses ##x## but then realizes that ##l## is actually in the direction of ##\vec F##, perpendicular to ##x##, and plays the role of ##y##. Using this information, they find the correct equation for ##l##, which is ##l = \frac{mu^2}{2eE}##.
Istiak
Homework Statement
An electron of charge e is moving at a constant velocity u, along x-axis. It enters a region of constant electric field E, which is pointing perpendicular to x-axis. The electron moves in a parabola. Which of the following represents the focal length of the parabola? Neglect any effects due to gravity. (for a parabola of type ##x^2=4ay##, a is the focal length)
Relevant Equations
##\int \vec F\cdot d\vec s = \frac{1}{2}mu^2##

Here I was going to use ##\int \vec F \cdot d\vec l = \frac{1}{2}mu^2##

What I got that is ##l=\frac{mu^2}{2eE}##. Here the question is what is ##l## (I took ##x## while doing the work but here I used ##l## instead of ##x##)? I was assuming that it's ##x## since I am calculating work in the parabola. So my equation stands ##a=\frac{m^2u^4}{16 y(eE)^2}##. But there's no option of it. But what I found for ##l## that satisfies. But my question is how ##l## is focal length?

##l## is in the direction of ##\vec F##, so perpendicular to your ##x##. It plays the role of ##y##, not ##x##

##\ ##

BvU said:
##l## is in the direction of ##\vec F##, so perpendicular to your ##x##. It plays the role of ##y##, not ##x##

##\ ##
So if I take ##y## into account then I get...

##y=\frac{mu^2}{2eE}##
##\frac{x^2}{4a}=\frac{mu^2}{2eE}##
##a=\frac{x^2 eE}{2a mu^2}##

but...

No you don't:
$$\left . \begin {array} {ll} x &= ut \\y & = \displaystyle {eEt^2\over 2m} \end{array}\right \}\Rightarrow y =x^2 {eE\over 2mu^2}\Rightarrow x^2 = 4\left(mu^2\over 2eE\right ) y$$so that $$a=\displaystyle {mu^2\over 2eE}$$

sorry I had to fumble with the 2's a few times...

##\ ##

Last edited:
Istiak

## 1. What is the focal length of an electron in a parabolic motion?

The focal length of an electron in a parabolic motion is the distance from the vertex of the parabola to the focus point, where the electron's path would converge if there were no external forces acting on it.

## 2. How is the focal length of an electron in a parabolic motion calculated?

The focal length of an electron in a parabolic motion can be calculated using the equation f = p/2, where f is the focal length and p is the distance between the vertex and the focus point.

## 3. What factors affect the focal length of an electron in a parabolic motion?

The focal length of an electron in a parabolic motion is affected by the initial velocity and angle of the electron, as well as any external forces such as electric or magnetic fields.

## 4. Can the focal length of an electron in a parabolic motion be changed?

Yes, the focal length of an electron in a parabolic motion can be changed by altering the initial conditions or by applying external forces to the electron's path.

## 5. Why is it important to find the focal length of an electron in a parabolic motion?

Finding the focal length of an electron in a parabolic motion can provide valuable information about the electron's trajectory and the forces acting on it. This can be useful in understanding and predicting the behavior of electrons in various systems and applications, such as in particle accelerators or electron microscopy.

Replies
1
Views
2K
Replies
1
Views
2K
Replies
2
Views
2K
Replies
3
Views
2K
Replies
5
Views
844
Replies
11
Views
603
Replies
36
Views
1K
Replies
3
Views
573
Replies
28
Views
858
Replies
15
Views
1K