In designing a velocity selector that uses uniform...

In designing a velocity selector that uses uniform perpendicular electric and magnetic fields, you want to select positive ions of charge $+5 e$ that are traveling perpendicular to the fields at 8.75 $\mathrm{km} / \mathrm{s}$ . The magnetic field available to you has a magnitude of 0.550 $\mathrm{T}$ (a) What magnitude of electric field do you need? (b) Show how the two fields should be oriented relative to each other and to the velocity of the ions. (c) Will your velocity selector also allow the following ions (having the same velocity as the $+5 e$ ions) to pass through undeflected: (i) negative ions of charge $-5 e,$ (ii) positive ions of charge different from $+5 e ?$

In designing a velocity selector that uses uniform perpendicular electric and magnetic fields, you want to select positive ions of charge $+5 e$ that are traveling perpendicular to the fields at 8.75 $\mathrm{km} / \mathrm{s}$ . The magnetic field available to you has a magnitude of
0.550 $\mathrm{T}$ (a) What magnitude of electric field do you need? (b) Show how the two fields should be oriented relative to each other and to the velocity of the ions. (c) Will your velocity selector also allow the following ions (having the same velocity as the $+5 e$ ions) to pass through undeflected: (i) negative ions of charge $-5 e,$ (ii) positive ions of charge different from $+5 e ?$

Key Concepts

Charge Dependence

The deflection of a charged particle in electric and magnetic fields depends on its charge magnitude and sign. The electric and magnetic forces both scale with the charge of the particle. Thus, only particles with the precisely required charge-to-mass ratio and velocity will experience perfectly balanced forces in a velocity selector, which means that variations in charge magnitude or sign will result in different deflections.

Velocity Selector

A velocity selector is a device that uses perpendicular electric and magnetic fields to allow charged particles of a specific velocity to pass through undeflected. This occurs when the electric force acting on a particle balances exactly with the magnetic force, resulting in no net deflection of particles that have the target speed.

Uniform Electric Field

A uniform electric field is a region in space where the electric field strength and direction remain constant. In the context of a velocity selector, the uniform electric field exerts a constant force on charged particles. Its magnitude is chosen such that when combined with the magnetic force, it cancels the net force on particles moving at the selected speed.

Uniform Magnetic Field

A uniform magnetic field has a constant magnitude and direction over a region of space. When a charged particle moves perpendicular to this field, it experiences a magnetic force that is proportional to its charge, velocity, and the magnetic field strength. This force acts perpendicular to both the velocity of the particle and the magnetic field.

Lorentz Force

The Lorentz force is the net force experienced by a charged particle moving in electric and magnetic fields. It is the vector sum of the electric force (qE) and the magnetic force (qvB). In a velocity selector, the condition for no deflection is achieved when these two forces are equal in magnitude but opposite in direction, thereby canceling each other.

Perpendicular Field Orientation

The perpendicular orientation of the electric and magnetic fields is crucial in a velocity selector. This arrangement ensures that the forces produced by each field act in opposite directions on the particle. Additionally, the particle’s velocity is perpendicular to both fields, which is necessary for the magnetic force to be maximized and to achieve the cancellation condition required for the selector to work.

Transcript

00:01 In this problem, we are working with both magnetic field forces as well as what forces due to electric fields.

00:08 In this problem, we want to design a velocity selector.

00:12 What this velocity selector does is it takes particles with a charge q is equal to 5e, e being the charge of a proton, and makes them move at a constant velocity, which we chose to be 8 .75 kilometers per second.

00:28 Now we chose a magnetic field we will use for this problem being 0 .55 teslas.

00:36 And what we need to do is figure out what does the electric field have to be in order to make this device work.

00:43 Now for this device to work, what we need is the force to be equal to zero when the velocity hits 8 .75 kilometers per second.

00:58 Right? because once we have this, that means that the particles won't be accelerating anymore once they hit this velocity.

01:06 Thus, they'll be moving at a constant velocity.

01:10 So our net force here is going to be the magnetic field force, which is qv cross b, plus the force due to the electric field, which is q times e.

01:25 So this is what we want to be zero.

01:28 That's the way we can do here.

01:30 It's just move, say, qe to one side.

01:35 So we get negative qe is equal to qv cross b.

01:42 If we could divide by our q's, they just go away, and we can take the magnitude of both sides.

01:51 What we can end up with is that e is equal to magnitude of the velocity times the magnitude of the magnetic fields.

02:01 And this, just plug it in the numbers, we get 4 ,812 volts per meter.

02:10 So this is an electric field that we need in order to make this device work...