(a) What value of magnetic field would make a beam of elec-...

(a) What value of magnetic field would make a beam of elec- trons, traveling to the right at a speed of $4.8 \times 10^{6} \mathrm{m} / \mathrm{s},$ go undeflected through a region where there is a uniform electric field of 8400 $\mathrm{V} / \mathrm{m}$ pointing vertically up? (b) What is the direction of the magnetic field if it is known to be perpendic- ular to the electric field? (c) What is the frequency of the circular orbit of the electrons if the electric field is turned off?

(a) What value of magnetic field would make a beam of elec-
trons, traveling to the right at a speed of $4.8 \times 10^{6} \mathrm{m} / \mathrm{s},$ go
undeflected through a region where there is a uniform electric
field of 8400 $\mathrm{V} / \mathrm{m}$ pointing vertically up? (b) What is the direction of the magnetic field if it is known to be perpendic-
ular to the electric field? (c) What is the frequency of the
circular orbit of the electrons if the electric field is turned off?

Key Concepts

Lorentz Force

This is the total force on a charged particle due to electric and magnetic fields. It is given by F = q(E + v × B), where the electric force (qE) and the magnetic force (q(v × B)) determine how the particle moves in combined fields.

Velocity Selector (Undeflected Motion Condition)

In devices such as velocity selectors, the electric and magnetic forces are balanced so that a charged particle experiences no net force. Setting qE equal to qvB ensures the particle travels in a straight line, which is used to select particles of a specific speed.

Right-Hand Rule for Force Direction

The direction of the magnetic force on a moving charge is determined by the right?hand rule applied to the cross product v × B. For electrons, which have negative charge, the force direction is opposite to that predicted by the standard right?hand rule, a key consideration when determining the orientation of the magnetic field relative to velocity and electric field.

Cyclotron Frequency

This concept describes the frequency at which a charged particle orbits in a uniform magnetic field. It is defined by the formula f = |q|B/(2?m) and is central to understanding the motion of particles in magnetic confinement applications such as cyclotrons.

Transcript

00:01 For this problem on the topic of magnetism, we want to know the value of the magnetic field that will make a beam of electrons which are moving to the right be undeflected as they travel through a region that is experiencing a uniform electric field.

00:15 We also want to know the direction of this magnetic field if it is particular to the electric field, as well as the frequency of the circular orbit of electrons if the electric field is then turned off.

00:28 Now for the beam of electrons to be undeflected, the magnitude of the magnetic force must equal to the magnitude of the electric force acting on the electrons.

00:37 So we'll assume that the magnetic field will be perpendicular to the velocity of the electrons so we can get the maximum magnetic force.

00:46 So if we equate the magnetic force fb to the electric force f e, this becomes qvb must equal to q times.

01:02 And from here we can see that the magnetic field strength b is simply equal to the electric field strength e over the speed of the electrons v and both of these values are known so the electric field strength is 8 ,400 volts per meter and the speed of the electrons is for 0 .8 times 10 to the power 6 meters per second and so this gives us the magnetic field strength to be 1 .75 times 10 to the minus 3 teslas.

01:47 So if we keep our significant figures, this is equal to 1 .8 times 10 to the minus 3 tesla's.

01:56 So that's the strength of the magnetic field that needs to exist in order to keep the electrons from being undeflected.

02:05 Or keep the electrons from being deflected rather.

02:09 Now in part b we want to know the direction of this magnetic field.

02:13 Now the electric field is pointing up and the electric force is down...

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