Charged Particles Moving in Magnetic Field

[Gear300]

The picture tube in a television uses magnetic deflection coils rather than electric deflection plates. Suppose an electron beam is accelerated through a 50.0 kV potential difference and then through a region of uniform magnetic field 1.00 cm wide. The screen is located 10.0 cm from the center of coils and is 50.0 cm wide. When the field is turned off, the electron beam hits the center of the screen. What field magnitude is necessary to deflect the beam to the side of the screen? Ignore relativistic corrections.

I'm thinking the coil extends to the screen...so it would be a coil being around 20 cm long and 1 cm wide. I can find the entrance velocity of the electron beam using the given voltage...after that point I'm stuck. Assuming the magnetic field is perpendicular to the velocity, the magnetic force should produce a centripetal motion that lasts only up until the particle moves 0.5 cm to the side from the central axis of the coil. From that point, the tangential velocity, which would have a magnitude equal to the one calculated using the voltage, would carry the particle to the side of the screen. I'm not sure of how to continue on from there.

 

[jbsteele]

Using the given information, we can calculate the velocity of the electron beam entering the magnetic field as follows:v = sqrt(2*50 000/9.11e-31) = 1.02e7 m/sThe magnetic force on a particle with charge q moving with velocity v in a magnetic field of magnitude B is given by F = qvB. In order to deflect the beam to the side of the screen, the magnetic force must be equal to the centripetal force on the particle as it moves in a curved path across the field. The centripetal force is given by F = mv^2/r, where m is the mass of the particle and r is the radius of curvature of its path. We can solve for B using these two equations. Since the distance from the center of the coils to the side of the screen is 10 cm (0.1 m) and the width of the field is 1 cm (0.01 m), the radius of curvature of the particle's path is 0.5 cm (0.005 m). Substituting these values into the equation for centripetal force, we get:F = mv^2/r = 9.11e-31 * (1.02e7)^2 / 0.005 = 1.45e-13 NSubstituting this value for F into the equation for the magnetic force, we get:F = qvB => B = F/qv = 1.45e-13 / (1.602e-19 * 1.02e7) = 1.42e-6 T

 

[Moetasim]

I would respond by first acknowledging that the scenario described is a classic example of the application of the Lorentz force law, which describes the force experienced by a charged particle moving in a magnetic field. The force is given by the equation F = qvB, where q is the charge of the particle, v is its velocity, and B is the magnetic field strength.

In this case, the electron beam is accelerated through a 50.0 kV potential difference, which gives it an initial velocity. As the beam enters the region of uniform magnetic field, it experiences a force that causes it to move in a circular path. This force is perpendicular to both the velocity of the electron beam and the magnetic field, resulting in the centripetal motion mentioned in the content.

To determine the necessary field magnitude to deflect the beam to the side of the screen, we can use the equation for centripetal force, F = mv^2/r, where m is the mass of the electron, v is its velocity, and r is the radius of the circular path. Since the beam is initially moving in a straight line, the radius of the circular path is equal to the distance from the center of the coils to the screen, which is 10.0 cm.

From the given information, we can calculate the initial velocity of the electron beam to be approximately 9.11 x 10^7 m/s. Plugging this value into the equation for centripetal force, and setting it equal to the Lorentz force, we can solve for the magnetic field strength. This calculation yields a field magnitude of approximately 0.005 T (tesla).

Therefore, in order to deflect the electron beam to the side of the screen, a magnetic field strength of 0.005 T is necessary. It is worth noting that this calculation assumes the electron beam is traveling perpendicular to the magnetic field, and that any deviations from this angle could affect the accuracy of the result.

In conclusion, the use of magnetic deflection coils in the picture tube of a television allows for precise control of the electron beam, resulting in the creation of the images we see on the screen. The application of scientific principles, such as the Lorentz force law, allows us to understand and manipulate the behavior of charged particles in magnetic fields.