How Does Lorentz Transformation Affect Electron Dynamics?

[Messy]

Homework Statement

An electron with rest mass m0 = 9.11E10-31 kg and charge e= 1.6E10-19 C was accelerated with an acceleration voltage U=30 kV

a) What is its velocity , momentum and energy in a classical picture ?
b)What is its velocity , momentum and energy in a relativistic picture ? What is its relativistic mass ?

c) If the acceleration voltage was only U=30 V , What is the electron's velocity , momentum and energy ? Would you use the classical or the relativistic picture and why ?

Homework Equations

For Part a) v = dx / dt
p = mv
energy = (1/2) m0 * v2

For Part b) Lorentz Transformation for Velocity
ux' = [ux - v] / [ 1+ (vux /c^2) ]
Where ux' as i understand is the velocity , moving frame of reference .

Relativisitc momentum
p= m0 * u / ((1-(u^2/c^2))^1/2

Relativistic Energy
E=mc^2

Relativisitic mass
mr = m0 / ((1-(u^2/c^2))^1/2

The Attempt at a Solution

If the above equations are correct in this case so

For part a) for velocity i m not getting how can i calculate the numerical figure
for momentum offcourse if i have velocity i can simply put it in the equation
for energy as well i need velcocity

For part b)

for velocity what is the difference in Ux and V ?
What U i have to substitute for relativistic momentum ?
For Energy :
mass = 9.11 E 10 -31
For Relativistic mass
m0 = 9.11 E 10 -31

Thanks in advance

 

[anathema]

for your help!



Thank you for your question. I will try my best to assist you in understanding the concepts involved in this problem. Please note that I will be using SI units in my explanation.

a) In a classical picture, we can use the equations you have provided to calculate the velocity, momentum, and energy of the electron.
Velocity: As you correctly stated, velocity is given by v = dx/dt. In this case, the acceleration voltage U is given in kilovolts (kV), so we need to convert it to volts (V) before using it in the equation. This can be done by multiplying U by 1000. So, the velocity of the electron can be calculated as follows:

v = √(2eU/m0) = √(2*1.6E-19*1000*30000/9.11E-31) = 5.84E7 m/s

Momentum: Momentum is given by p = mv. We have already calculated the velocity in the previous step, so we can use it to calculate the momentum as follows:

p = m0v = 9.11E-31*5.84E7 = 5.32E-23 kg*m/s

Energy: Energy is given by (1/2)m0v^2. We already have the values for m0 and v, so we can use them to calculate the energy as follows:

energy = (1/2)m0v^2 = (1/2)*9.11E-31*5.84E7^2 = 1.62E-12 J

b) In a relativistic picture, we need to use the equations you have provided to calculate the velocity, momentum, energy, and relativistic mass of the electron.

Velocity: The Lorentz transformation for velocity is used to calculate the velocity in a different frame of reference. In this case, we are calculating the velocity in the rest frame of the electron. So, we can use the following equation:

ux' = [ux - v] / [ 1+ (vux /c^2) ]

Here, ux is the velocity of the electron in the rest frame, and v is the velocity of the rest frame. In this case, the rest frame is at rest, so v = 0. Substituting the values, we get:

ux

 

[thomate1]

.
I would like to clarify that the Lorentz Transformation is a mathematical tool used in the theory of special relativity to describe how measurements of space and time change for an observer in one frame of reference compared to another moving frame of reference. It is not an equation or formula in itself, but rather a set of equations that allow us to transform coordinates and measurements between different frames of reference.

In regards to the given homework statement, I would suggest starting by understanding the concepts of velocity, momentum, and energy in both classical and relativistic physics. In classical physics, velocity is simply the rate of change of position, momentum is the product of mass and velocity, and energy is the kinetic energy given by the equation (1/2) m0 * v^2. In relativistic physics, the equations for velocity, momentum, and energy are different due to the effects of special relativity.

For part a), you can use the classical equations to calculate the velocity, momentum, and energy of the electron. Simply plug in the given values for mass, charge, and acceleration voltage in the equations provided in the homework statement.

For part b), you will need to use the Lorentz Transformation equations to calculate the velocity, momentum, and energy in a relativistic picture. The equation for velocity is the one you have provided, where u is the velocity of the electron in the original frame of reference and u' is the velocity in the moving frame of reference. For relativistic momentum, you can use the equation provided in the homework statement, where u is the velocity of the electron and m0 is the rest mass. For relativistic energy, you can use the equation E=mc^2, where m is the relativistic mass of the electron, given by mr = m0 / ((1-(u^2/c^2))^1/2.

For part c), if the acceleration voltage is only 30 V, the electron will not have a significant change in its velocity, momentum, or energy. In this case, you can use the classical equations to calculate these values as the effects of special relativity will be negligible.

I hope this helps clarify the concept of Lorentz Transformation and how it can be applied to solve the given homework statement. Remember to always understand the fundamental concepts before applying equations and to double-check your calculations to ensure accuracy.