Electromagnetism - Lorentz Force

In summary: E/mIn summary, the particle moves in a circle that is a cycloid as a result of the electric and magnetic fields.
  • #1
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Homework Statement



A particle of charge q starts from rest at the origin of coordinates in a region where there is a uniform electric field of strenth E parallel to the x-axis, and a uniform magnetic field B parallel to the z-axis.

Find the equations of motion, and solve them to show that the coordinates of the particle at a time t later will be:

x = (E/B*omega)*(1 - cos(omega*t))

y = - (E/B*omega)*(omega*t - sin(omega*t))

z = 0

where omega = q*B/m. (The path of the circle is a cycloid.)

Homework Equations



The parametric equation of a cycloid:

x = constant*(1 - cos(omega*t))

y = constant*(omega*t - sin(omega*t))

The force acting on the particle:

F = q*E + q*vxB

The Attempt at a Solution



I've done some work on this problem and so far the equations of motion that I've got for the particle are as follows:

1) F(x) = q*E + q*v(y)*B -> x[double-dot] = q*E/m + omega*y[dot]

2) F(y) = -q*v(x)*B -> y[double-dot] = -q*B*x[dot]

I've tried integrating these equations once (eg. integrate 2)) and then substituting this into the other equation. This then gave me:

x[double-dot] + omega^2*x = E*B

And this is where I'm stuck. This has the form of a simple harmonic oscillator, except that the r.h.s. isn't zero, so I can't solve it. Also, I'm not even sure if everything that I've done so far is correct.

Any help on this would be very much appreciated!
 
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  • #2
I found the same equations of motion but i kinda stuck there as well. I`ll check it again tomorrow :)
 
  • #3
Thanks, that would be really helpful! I'm glad to know that I don't seem to be a complete moron though, or that I've overlooked some vital detail that will make the whole thing ridiculously easy.
 
  • #4
Solving the differential equation x[double-dot]+x*ω^2=qE/m (1)

X=X1+X2=(c1*cosωt+c2*sinωt)+qE/(m*ω^2) , c1,c2=constants

X1 is the solution of the : x[double-dot]+x*ω^2=0
X2 is a "special" solution that satisfies (1).

I think you can solve that now.
x(0)=0
x[dot](0)=0
 
Last edited:

1. What is the Lorentz force?

The Lorentz force is a fundamental force of nature that describes the interaction between electric and magnetic fields. It is responsible for the movement and behavior of charged particles in the presence of these fields.

2. How does the Lorentz force affect charged particles?

The Lorentz force causes charged particles to experience a force perpendicular to both the direction of their motion and the direction of the magnetic field. This force can cause the particles to change direction or speed up.

3. What is the equation for the Lorentz force?

The equation for the Lorentz force is F = q(E + v x B), where F is the force, q is the charge of the particle, E is the electric field, v is the velocity of the particle, and B is the magnetic field.

4. How is the Lorentz force used in everyday life?

The Lorentz force is used in a variety of everyday technologies, such as electric motors, generators, and particle accelerators. It is also used in medical imaging techniques and in electronic devices like speakers and headphones.

5. What is the relationship between the Lorentz force and the electromagnetic force?

The Lorentz force is a component of the larger electromagnetic force, which also includes the electric force and the magnetic force. The electromagnetic force is responsible for many phenomena, including the behavior of magnets, the properties of light, and the structure of atoms.

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