Relativistic Transformation of Lorentz Force (E + v x B)

In summary, the Lorentz force is not invariant between frames, which can lead to confusion when trying to calculate the force on a test charge.
  • #1
mairzydoats
35
3
Here is a problem:
Imagine two equally charged capacitor plates parallel to the x-y axis, whose area is large enough compared to the distance between them that fringe effects can be ignored. The bottom plate (at z=0) is + charged, and the top is - charged. The vector field E is therefore directed upward ('upward' in this case being the positive 'z' direction) from the + plate to the - plate, and its magnitude should be equal to [surface charge density/epsilon].

If we examine the Lorentz force q[E + vxB] experienced by a test charge situated between the two plates, and moving with constant velocity parallel to the plates as seen from the frame of reference in which the plates are at rest, relativistic theory tells us we should get the same total Lorentz force (albiet with different E and B values) when we examine it in the frame of the moving charge, in which the test charge is still, and the plates are in motion.

Well then, first let's look at the frame where the plates are at rest and the test charge moves. Because the plates are at rest, B must be zero, and therefore the vxB term must also be zero. The Lorentz force reduces to q[E].

But now let's look at the frame of the moving test charge. In this frame, there is a B-field due to the moving plates, but because we are moving with the test charge, it is at rest in this frame ... and therefore magnitude of B doesn't effect the force it experiences. The vxB term once again vanishes, leaving the Lorentz force to be q[E'], where E' means the electric field of the first reference [E] frame as transformed by the lorentz factor of [1/(1-v squared/c squared)].

But q[E] does NOT equal q[E']!

What is wrong with this analysis?
 
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  • #2
mairzydoats said:
we should get the same total Lorentz force
The Lorentz force is a three vector, so it is not invariant.
 
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  • #3
DaleSpam said:
The Lorentz force is a three vector, so it is not invariant.
Thank you for your reply. I apologize if my understanding of the subject matter is not advanced enough for me to know what you mean. How can the force on something, which is a 'real event', change simply because change our inertial frame of obervation?
 
  • #4
mairzydoats said:
Thank you for your reply. I apologize if my understanding of the subject matter is not advanced enough for me to know what you mean. How can the force on something, which is a 'real event', change simply because change our inertial frame of obervation?
Force is mass times acceleration, and acceleration is the second derivative of distance with respect to time. Due to time dilation and length contraction acceleration in one frame is different from acceleration in another frame, therefore so is force.
 
  • #5
DaleSpam said:
Force is mass times acceleration, and acceleration is the second derivative of distance with respect to time. Due to time dilation and length contraction acceleration in one frame is different from acceleration in another frame, therefore so is force.

but the force in my question is created by an E field whose only component is in the positive z-direction, and this, along with any acceleration it may create, is also in the positive z-direction. The transformation of these frames parallel to the x-y plane the test charge moves in doesn't contract length parallel to z, correct?
 
  • #6
mairzydoats said:
but the force in my question is created by an E field whose only component is in the positive z-direction, and this, along with any acceleration it may create, is also in the positive z-direction. The transformation of these frames parallel to the x-y plane the test charge moves in doesn't contract length parallel to z, correct?

oh wait a minute ... [duhh] ... time still dilates. ha-ha. I guess a clock moving with my test charge would tick less than a clock situated on one of the plates as the charge is accelerated in the positive z direction by either E or E'. I still really have to mull this over & study before I'm quite comfortable. But thanks for pointing me in the right direction!
 

1. What is the Relativistic Transformation of Lorentz Force?

The Relativistic Transformation of Lorentz Force is a mathematical equation that describes the relationship between electric and magnetic fields in a moving frame of reference. It takes into account the effects of special relativity, such as time dilation and length contraction, on the force experienced by a charged particle in a moving magnetic field.

2. How is the Relativistic Transformation of Lorentz Force derived?

The Relativistic Transformation of Lorentz Force is derived from the Lorentz force law, which describes the force experienced by a charged particle in an electric and magnetic field. By applying the principles of special relativity, including the Lorentz transformation equations, the equation for the transformed force can be obtained.

3. What is the difference between the Relativistic Transformation of Lorentz Force and the classical Lorentz force?

The classical Lorentz force only takes into account the effects of electric and magnetic fields on a stationary charged particle. The Relativistic Transformation of Lorentz Force also considers the effects of special relativity, such as the particle's velocity and the observer's frame of reference, resulting in a more accurate and comprehensive representation of the force.

4. How does the Relativistic Transformation of Lorentz Force affect the motion of a charged particle?

The Relativistic Transformation of Lorentz Force can affect the motion of a charged particle in a number of ways. For example, it can cause time dilation and length contraction, resulting in changes to the particle's perceived velocity and trajectory. It can also affect the amount of work done on the particle and the amount of energy it gains or loses.

5. What are some practical applications of the Relativistic Transformation of Lorentz Force?

The Relativistic Transformation of Lorentz Force is used in a wide range of fields, including particle physics, astrophysics, and engineering. It is crucial for understanding the behavior of particles in accelerators, as well as the interactions between charged particles in magnetic fields. It is also used in the development of advanced technologies, such as particle accelerators and magnetic propulsion systems.

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