Particle released into electric and magnetic fields perpendicular

[mathnerd15]

Homework Statement

z+ up with E field, y to the right, x out of the page with B field
particle released from rest at (0,0,0) with only initial y velocity y=(E/B)t
or initial y=(E/2B)t

Homework Equations

can you suggest a good differential equations text?
y(t)=C1cos(wt)+C2sinwt+(E/B)t+C3
z(t)=C2cos(wt)-C1sin(wt)+C4

The Attempt at a Solution

so particle will move in a linear straight line to the right?

in the case of particle released from rest it will move in perfect half circle paths to the right? so there is a preservation of the geometry?

 

[mfb]

so particle will move in a linear straight line to the right?

Only for special values of E and B.

In general, you can calculate those coefficients C1 to C4 in your equations (they are not differential equations, by the way). Alternatively, you can derive those equations from the differential equation given by the Lorentz force.

in the case of particle released from rest it will move in perfect half circle paths to the right?

Right/left depends on the sign of E and B.

so there is a preservation of the geometry?

What does that mean?

 

[mathnerd15]

thanks very much! the geometry of the E and B fields/forces is perpendicular and coplanar and are simply rotated pi/2 clockwise

so...how do you sketch the graphs of these equations for the case?
v(0)=(E/2B)y, v(0)=(E/B)(y+z), I can find the centers of the circles and differentiate to find min/max but as sinusoidal functions they have some vague wave graph

do I just plug in values for 0, pi/2, pi, 3pi/2...?

 

[mfb]

do I just plug in values for 0, pi/2, pi, 3pi/2...?

That is a good idea, indeed.

 

[paranormal]

Yes, the particle will move in a straight line to the right in the y-direction due to the initial y-velocity. However, the presence of both an electric and magnetic field will cause the particle to also experience a force in the z-direction due to the electric field and a force in the x-direction due to the magnetic field. This will result in the particle following a curved path, rather than a straight line.

In the case of a particle released from rest, it will indeed move in a half-circle path to the right. This is because the initial y-velocity is equal to the electric field divided by the magnetic field (y=(E/B)t), which will result in a circular motion. This is known as the Lorentz force, where the force on a charged particle in an electric and magnetic field is given by F=q(E+vxB).

As for a good differential equations text, it would depend on your level of understanding and what specific topics you are interested in. Some popular texts for introductory differential equations include "Elementary Differential Equations" by Boyce and DiPrima and "Differential Equations with Boundary-Value Problems" by Zill and Cullen. For more advanced topics, "Partial Differential Equations for Scientists and Engineers" by Farlow and "Partial Differential Equations: An Introduction" by Strauss are good options.