Video: Recalling the Effect of Electric and Magnetic Fields on Moving Charges

Which of the following can deflect a beam of electrons? I) An electric field II) A nearby permanent magnet III) A magnetic field [A] II only [B] I only [C] II and III only [D] I, II, and III [E] III only

05:30

Video Transcript

Which of the following can deflect a beam of electrons? I) an electric field, II) a nearby permanent magnet, III) a magnetic field. a) II only, b) I only, c) II and III only, d) I, II, and III, e) III only.

The question asks us to identify which of the choices could deflect a beam of electrons. So let’s first identify what a beam of electrons is and then figure out how one could deflect. Electrons, usually given the symbol e minus, are negatively charged subatomic particles. An electron beam is just a collection of electrons travelling in a similar direction with similar velocities. In the diagram, the electrons, shown as little circles, have similar trajectories in that they’re all moving towards the right and staying within the confines of the dotted lines.

An electron beam deflects when it changes direction. Thus, in this continuation of our drawing of the beam, the area marked deflection shows a change in direction from the initial trajectory off to the right to a final trajectory off to the bottom right. A deflection is a change in direction, but a change in direction is only caused by acceleration. But objects only accelerate when forces act on. So connecting these two, an electron beam will only deflect when a force acts on it. In other words, deflection results from applied force. So the correct answer will identify all of the choices that can apply a force to a beam of electrons.

Remember that electrons are charged particles. So let’s recall two ways that charged particles can experience a force. The two ways are in the presence of an electric field, an 𝐸-field, or a magnetic field, a 𝐵-field. In an electric field, a particle would experience a force proportional to its charge and the strength of the local electric field. Symbolically, we write 𝐹, the force, is equal to 𝑞, the charge, times 𝐸, the electric field. This is true regardless of whether or not the particle is moving. So a series of electrons moving in an electron beam in the presence of an electric field would feel an applied force, and the beam could deflect. Therefore, option number I, an electric field, is a possible source of electron beam deflection. Therefore, choices a), c), and e) are not correct since they don’t include option I.

What about a magnetic field? A particle in a magnetic field fuels a force according to the following equation. 𝐹, the force, is equal to 𝑞, the charge, times the quantity, velocity 𝑣, cross magnetic field 𝐵. Where we’ve emphasized the vector nature of velocity and magnetic field since we’re forming a vector product, specifically the cross product. The important thing to gather from this equation is that a charged particle can experience a force from a magnetic field, provided the velocity is not zero. In other words, it’s moving, and the velocity and the magnetic field are not in the same direction. Since if they are, the cross product will be zero. This is in contrast to the electric field on a charged particle, which is nonzero as long as the electric field is nonzero.

Remember, though, that an electron beam fundamentally involves electrons that are moving. In other words, their velocities are nonzero. Therefore, an electron beam in a magnetic field will experience a force as long as the beam is moving along a different direction than the field. And so, choice III, a magnetic field, is indeed a possible source of electron beam deflection. At this point, we can already rule out choice b), I only, since option III is also correct. This leaves us with choice d) — I, II, and III — as the correct answer.

Let’s confirm that option II, a nearby permanent magnet, could indeed deflect a beam of electrons. Here, we’ve drawn a simple permanent bar magnet, with the north pole labelled N and the south pole labelled S. Now, we’ve added some representative field lines and their associated directions to the drawing. Now, let’s introduce an electron beam coming in from the left and travelling with a trajectory towards the right. Clearly, this electron beam is travelling in the presence of a magnetic field whose direction is not the same as its velocity. And so, therefore, this electron beam would indeed deflect. This confirms that option number II, a nearby permanent magnet, could indeed cause an electron beam to deflect. To conclude, let’s discuss the circumstances under which an electric field would indeed cause an electron beam to deflect.

Say, as usual, we have an electron beam travelling off towards the right. Now let this beam be in the presence of an electric field pointing in the opposite direction. In this case, the electric force, 𝑞𝐸, would act in the same direction as the beam, directly off to the right. Since the resulting force points in the same direction as the velocity, the electrons would only speed up. They wouldn’t change direction. The same would be true if the electric field reverse direction, so they’re pointed in the same direction as the beam. Only this time the electrons would slow down. But, again, they wouldn’t change direction.

On the other hand, if in this picture the electric field pointed downward, the resulting force would point upward. And the resulting beam would deflect upward. Now, if we swapped the direction of the electric field so that the electric field points up, the force points down. And the beam deflects down. Therefore, just like a magnetic field, an electric field will only deflect a beam of electrons if the electric field and the velocity of the beam are not parallel. The difference between these two cases is that if the velocity is parallel to the field in the magnetic case, there is no force. But in the electric case, there is a force. It just doesn’t deflect the beam.