Cathode Ray Tube in a uniform magnetic field

[mad_scientist]

Homework Statement

A CRT is placed in a uniform magnetic field B, with the axis of the tube parallel to the lines of force. If electrons emerging from the gun with a velocity v make an angle θ as they pass through the origin O so that their trajectory is a helix, show that a) they will touch the axis again at the time t= 2πm/Bq
b) that the coordinate of the point of touching is x= 2π(mv/Bq) (cosθ)

Homework Equations

t= 2πm/Bq
x= 2π(mv/Bq) (cosθ)

The Attempt at a Solution

 

[jbsteele]

a) The time taken for an electron to travel from the origin O to a point on the axis is given by 2πm/Bq.b) The displacement of the electron from O to the point on the axis is given by x= 2π(mv/Bq) (cosθ).

 

[tomcue]

I would approach this problem by first understanding the basic principles involved. A cathode ray tube is a device that uses a beam of electrons to create an image on a fluorescent screen. The electrons are accelerated by an electric field and then deflected by a magnetic field, allowing them to be focused and directed to specific points on the screen. In this case, the tube is placed in a uniform magnetic field, which means that the strength and direction of the field are constant throughout the region.

Based on this setup, we can use the equations provided to determine the time and coordinates at which the electrons will touch the axis again. The equation t=2πm/Bq represents the time it takes for an electron with charge q and mass m to complete one full revolution in a magnetic field B. This is known as the cyclotron frequency. Therefore, when the electron touches the axis again, it will have completed one full revolution.

Similarly, the equation x=2π(mv/Bq)(cosθ) represents the distance along the x-axis at which the electron will touch the axis again. This equation takes into account the velocity of the electron (v), the strength of the magnetic field (B), and the angle (θ) at which the electron passes through the origin O.

To show that these equations are correct, we can use the known values for the mass and charge of an electron, as well as the given velocity and angle, to calculate the expected time and coordinate of touching. By plugging in these values, we should arrive at the same results as the equations provided.

In conclusion, the equations t=2πm/Bq and x=2π(mv/Bq)(cosθ) accurately represent the time and coordinate at which an electron in a CRT will touch the axis again in a uniform magnetic field. This understanding can be applied to further study and improve the design and functionality of cathode ray tubes and other devices that use magnetic fields to manipulate electron beams.