Particle moving relativistically through E field

[Liwk]

Homework Statement

Hi.
This is the problem I'm trying to solve:

Consider the trajectory of a charged particle moving with a speed 0.8c in the x direction when it enters a large region in which there is a uniform electric field in the y direction. Show that the x velocity of the particle must actually decrease. What about the x component of momentum?


2. The attempt at a solution and Relevant Equations
This one is for discussion mainly. Thus, these are my conclusions:

When the charged particle enters on the Electric Field, F=qE is perpendicular to the Px-component of the momentum (as it is in y-direction). Thus, there's no force acting on the Px-component of the motion. So, we can use Momentum Conservation Principle.

This is,

Px_i= Px_f

m₀Vx_i/(1-(Vx_i²/c²))^1/2 = m₀Vx_f/(1-(Vx_f²+Vy²)/(c²))^1/2
​

Solving for Vx_f,

Vx_f² = Vx_i²(1-(Vy/c)²)​

Thus if Vy increases, then Vx_f decreases.

Am I correct?

 

[anathema]

Your analysis and conclusion are both correct. When a charged particle enters a uniform electric field, it will experience a force perpendicular to its velocity, causing it to change direction. This means that the x component of its velocity will decrease, while the y component will increase. This is in accordance with the principle of conservation of momentum, which states that the total momentum of a system must remain constant in the absence of external forces.

Your calculation of the final x velocity using the conservation of momentum equation is also correct. As you have shown, if the y velocity increases, the x velocity must decrease in order to conserve momentum. This is because the total momentum of the particle must remain constant, and increasing the y velocity means there must be a corresponding decrease in the x velocity to keep the total momentum the same.

I hope this helps with your problem. Keep up the good work in your studies!



Scientist

 

[thomate1]

I can confirm that your reasoning and conclusion are correct. When a charged particle enters a region with a uniform electric field, the electric force acts perpendicular to the direction of motion, so it does not affect the x velocity of the particle. However, the electric force does affect the y velocity, causing it to increase. This results in a decrease in the x component of momentum, as you correctly stated. This phenomenon is known as relativistic momentum and is described by the equation you used. Good job!