Video Transcript
A conducting rod moves through a
uniform magnetic field, as shown in the diagram. Which end of the rod has a higher
potential?
To begin, we should recall that the
electric potential at a point helps tell us the amount of work it would take to move
a test point charge, that is, a single positively charged particle, to that
point. Generally, electric potential is
greater near a positive charge than it is near a negative charge, since it takes
more work to push a positive test charge closer to a positive charge than it would
to push a positive test charge closer to a negative charge.
So, what does this have to do with
a rod moving through a magnetic field? Well, we should also recall that
when a straight conductor moves through a uniform magnetic field, there is an
electromotive force, or emf, induced across it with a magnitude of 𝑙𝑣𝐵 sin 𝜃,
where 𝑙 is the length of the conductor, 𝑣 is its speed, 𝐵 is the strength of the
magnetic field, and 𝜃 is the angle between the conductor’s velocity and the
magnetic field.
In this question, we have a rod
that’s moving to the right through a magnetic field that points into the screen. So the rod’s direction of motion is
perpendicular to the magnetic field, and 𝜃 equals 90 degrees. We know that the sin of 90 degrees
just equals one. So here, the formula for the
induced emf simplifies to 𝑙𝑣𝐵. Now, the length of the rod, its
speed, and the strength of the magnetic field are all nonzero. So we know that there will indeed
be an induced emf across this rod.
Thus, as the rod moves through the
field, free charges in the rod will be accelerated toward the rod’s top or
bottom. This buildup of negative charge at
one end and positive charge at the other end constitutes a potential difference
across the rod. So, if we can just determine the
sign of the charge at either end, we can compare the electric potential at either
end and answer this question. To do this, let’s first determine
the magnetic force that acts on free electrons in the rod using the right-hand
rule.
To determine the force on a
particle in the rod, we need to first identify the direction of 𝑞𝑣, where 𝑞 is
the charge of the particle and 𝑣 is its velocity. We know that 𝑣 alone points to the
right. But since we’re presently thinking
about the force on an electron, this 𝑞-term introduces a negative sign to 𝑞𝑣. And so 𝑞𝑣 points left. Now, using our right hand, we point
our fingers in the direction of 𝑞𝑣 and then curl them in the direction of the
magnetic field. Here, the magnetic field points
into the screen. So we have to flip our hand upside
down in order to curl our fingers that way. Then, the thumb points in the
direction of the force on the particle.
Since our right hand has to make a
thumbs-down shape here, we know that free electrons in the wire experience a
magnetic force that accelerates them downward. This means that as the rod moves,
negative charge will build up at the bottom end of the rod, B. And likewise relative positive
charge will build up at the top end of the rod, A.
Finally, we just need to compare
the electric potential at the two ends of the rod. Since electric potential is
generally greater near positive charge than it is near negative charge, we know that
the end labeled A has a higher potential. So, when asked whether end A or B
of this rod has a higher potential, we know that the answer is A.
