Minimum Potential Difference Required

[E&M]

Homework Statement

Consider a magnetron. It consists of an electron-emitting filament at the center of a cylindrical anode situated in a uniform magnetic field. Electrons of charge e and mass m are emitted with negligible velocity from the filament.

What is the minimum potential difference between the filament and anode for elctrons to reach the anode.

Homework Equations

PE = KE

F = d (mv)/dt

The Attempt at a Solution

total kinetic energy of the electron = radial KE + Tangential KE
Total KE = PE of the electron (i.e. 'eV')

 

[Redbelly98]

I'm not sure, but it looks like you'll need to write out expressions for the force (due to electric and magnetic fields), and solve the differential equation for the electron's motion. This does not look trivial.

Not sure if conservation-of-energy can be applied here, since the magnetic field causes the electron paths to curve.

 

[thomate1]

= (1/2)mv^2 + (1/2)mv^2
= mv^2

The minimum potential difference required for electrons to reach the anode can be calculated using the equation PE = KE. This means that the potential energy (PE) of the electron must be equal to its kinetic energy (KE) in order for it to reach the anode. Therefore, the minimum potential difference required would be equal to the kinetic energy of the electron, which can be calculated using the formula KE = (1/2)mv^2, where m is the mass of the electron and v is its velocity.

In order to determine the velocity of the electron, we can use the equation F = d (mv)/dt, where F is the force acting on the electron, d is the distance between the filament and the anode, and dt is the time it takes for the electron to travel this distance. The force acting on the electron is given by the Lorentz force, which is equal to e(v x B), where e is the charge of the electron, v is its velocity, and B is the magnetic field.

Combining these equations, we can determine the minimum potential difference required by setting the potential energy equal to the kinetic energy and solving for v. The resulting equation is v = (2eBd/m)^(1/2), where B is the magnetic field and d is the distance between the filament and the anode. This means that the minimum potential difference required is dependent on the magnetic field and the distance between the filament and the anode.

In conclusion, the minimum potential difference required for electrons to reach the anode in a magnetron can be calculated using the equation v = (2eBd/m)^(1/2), where B is the magnetic field and d is the distance between the filament and the anode. This potential difference is necessary for the electrons to have enough kinetic energy to overcome the force of the magnetic field and reach the anode.