# Charge Trajectory

1. Oct 4, 2015

### UMath1

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?

2. Oct 4, 2015

### UMath1

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?

3. Oct 4, 2015

### Staff: Mentor

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?

4. Oct 4, 2015

### UMath1

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

I am not familiar with the lorentz force.

5. Oct 4, 2015

### Staff: Mentor

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.

6. Oct 4, 2015

### UMath1

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

7. Oct 4, 2015

### Staff: 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?

8. Oct 4, 2015

Yes

9. Oct 4, 2015

### UMath1

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

10. Oct 4, 2015

### Staff: Mentor

Now that you've seen the definition of the Lorentz force, what do you think? F=ma, right?

11. Oct 4, 2015

### UMath1

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.

12. Oct 4, 2015

### Staff: Mentor

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.

13. Oct 4, 2015

### Staff: 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.
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.

14. Oct 4, 2015

### Staff: Mentor

Yes, sorry. In his initial example, I thought he was talking about straight E-field lines.

15. Oct 5, 2015

### UMath1

So how would it move?

16. Oct 5, 2015

### ZapperZ

Staff Emeritus
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.

17. Oct 5, 2015

### UMath1

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.

18. Oct 5, 2015

### ZapperZ

Staff Emeritus
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.

19. Oct 5, 2015

### UMath1

yes

20. Oct 5, 2015

### ZapperZ

Staff Emeritus
Fine.

Now tell me exactly the field lines that you have.

Zz.