# Charge Trajectory

My textbook said that field lines do not always represent the trajectory of a charged particle in a field. How would you find the trajectory then?

## Answers and Replies

What points? Lets say you had a field created by one positive charge and one negative charge. If a negative test charge would not flow along the field line from positive to negative, how would it flow if it started at the positive end?

berkeman
Mentor
My textbook said that field lines do not always represent the trajectory of a charged particle in a field. How would you find the trajectory then?
What field? Electric or Magnetic or both? A static field or a changing field? What it the initial trajectory of the charged particle with respect to the field(s)? Are you familiar with the Lorentz Force? Can you show us that equation?

Electrostatic field. The negatively charged particle is just let go from the positive end.

I am not familiar with the lorentz force.

berkeman
Mentor
Electrostatic field. The negatively charged particle is just let go from the positive end.

I am not familiar with the lorentz force.
The Lorentz Force is the vector force acting on a charged particle that is moving in magnetic and/or Electric fields:

$$F = qE + q(vxB)$$

If you just release a charged particle from rest, yes it will follow the E-field lines, because that's the direction of the Lorentz force when there is no B field.

So only if it has an initial velocity, it won't follow the field lines?

berkeman
Mentor
If you shot the charged particle into a region of E-filed that was perpendicular to the motion, then the particle would follow a curved path, correct?

Yes

berkeman
but if you shot it in the direction of a field line, then?

berkeman
Mentor
but if you shot it in the direction of a field line, then?
Now that you've seen the definition of the Lorentz force, what do you think? F=ma, right?

Right, but the direction of the force constantly varies if the field line is curved as it often is between positive and negative charges. It should move tangent to the field line initially. But I am not sure if after it moves in that direction, it will still be acted on by the same field line.

Dale
Mentor
2020 Award
To find the equation of motion just write down F=ma for the system and solve. Alternatively you can use the Lagrangian to determine the equation of motion.

mfb
Mentor
If you just release a charged particle from rest, yes it will follow the E-field lines, because that's the direction of the Lorentz force when there is no B field.
It will do that "in the first moment", but it won't do that in general, even if it starts at rest. If the field lines are curved, the particle won't follow them.
Right, but the direction of the force constantly varies if the field line is curved as it often is between positive and negative charges. It should move tangent to the field line initially. But I am not sure if after it moves in that direction, it will still be acted on by the same field line.
Your doubt is right here. To follow a curved line it would need an acceleration component orthogonal to the field line, but that does not exist by definition of the field line.

A magnetic field can change the trajectory, but in general it won't make particles follow any field lines.

berkeman
Mentor
It will do that "in the first moment", but it won't do that in general, even if it starts at rest. If the field lines are curved, the particle won't follow them.
Yes, sorry. In his initial example, I thought he was talking about straight E-field lines.

Your doubt is right here. To follow a curved line it would need an acceleration component orthogonal to the field line, but that does not exist by definition of the field line.
So how would it move?

ZapperZ
Staff Emeritus
So how would it move?
Have you solved a projectile motion in gravity? I find it hard to believe that you are doing charged particle in electric field and not have already tackled a projectile motion. If you have, then you really should go back and look at it, because, believe it or not, you already know how to solve this.

If you look at the projectile motion, the "field lines" are always pointing down, because this is the field lines for gravity. If you simple let go of a mass, how does it move? Does it move in the same direction of the "field lines"?

But what if you shoot the mass out at some angle with respect to the horizontal? How does it move then?

Look at the electrostatic problem and see why it is no different.

Zz.

But the field lines for gravity point in a constant direction. When you have a positive and negative charge, the field lines take a curved path. If a charge carrier is released in the direction of one of these curved lines, a force tangent to the curved line will act on it causing acceleration tangent to the line. But after that I don't see where the force will act on it.

ZapperZ
Staff Emeritus
But the field lines for gravity point in a constant direction. When you have a positive and negative charge, the field lines take a curved path. If a charge carrier is released in the direction of one of these curved lines, a force tangent to the curved line will act on it causing acceleration tangent to the line. But after that I don't see where the force will act on it.
So let me get this clear. If I give you an electrostatic field that is constant everywhere, and looks just like a gravity field, you can solve this easily, correct?

Zz.

yes

ZapperZ
Staff Emeritus
yes
Fine.

Now tell me exactly the field lines that you have.

Zz.

Lets say I release a negative charge from the top of the blue charge.

ZapperZ
Staff Emeritus
Lets say I release a negative charge from the top of the blue charge.
View attachment 89785
I don't understand what you mean by "... from the top of the blue charge..."

Regardless of that, when you have a field such as this, for an electron, the trajectory will closely follow the field lines when you release the charge from rest. However, the larger the mass, the longer the path, and the more accurate you want your answer, the more it will deviate from these field lines. If you want an accurate modeling of such path, you will have to solve the equation of motion for such a system, and it may not be an analytical solution (i.e. you might only be able to solve it numerically).

It is why you were never asked to solve for the path of a space craft going through the solar system when you dealt with gravity.

Zz.

mfb
Mentor
Regardless of that, when you have a field such as this, for an electron, the trajectory will closely follow the field lines when you release the charge from rest.
In general it will not.
Using the position description of post #21 and some rough estimate on the positions, I would expect our test mass to be unbound. It won't follow the field lines - it will not even stay in the picture, but escape nearly vertically, crossing field lines nearly orthogonally on its way out.

To make the charged particle follow field lines, you would need some resistance (like residual gas for collisions).

The mass is not relevant here, it just changes the timescale, but not the result.