Video Transcript
A conducting rod moves on
conducting rails that form a circuit that contains two resistors, as shown in the
diagram. The power dissipated by the circuit
is 65.5 milliwatts. The strength of the magnetic field
the circuit is within is 945 milliteslas. The rod has a resistance per unit
length of 15 ohms per meter. Find the speed 𝑣 at which the rod
must move.
Let’s start by taking a closer look
at this circuit. We see two resistors with known
resistance values. So let’s label them as 𝑅 one and
𝑅 two. But this circuit actually consists
of three resistors, because the rod also has some resistance whose value we don’t
currently know but we will be able to calculate. For now, let’s just call the
resistance of the rod 𝑅 sub 𝑟. Notice that there’s no symbol
showing a cell in this circuit, and that’s okay.
Recall that when a straight
conducting rod moves through a uniform magnetic field, an electromotive force, or
emf, is induced across the rod with a magnitude of 𝑙𝑣𝐵 sin 𝜃, where 𝑙 is the
length of the rod, 𝑣 is the speed of the rod, 𝐵 is the strength of the magnetic
field, and 𝜃 is the angle between the rod’s velocity and the magnetic field.
In this question, the rod is moving
to the right and the magnetic field points out of the screen. So the rod’s velocity is
perpendicular to the magnetic field, and 𝜃 equals 90 degrees. We know that the sin
of 90 degrees simply equals one, which means that an emf of magnitude 𝑙𝑣𝐵 will be
induced across the rod as it moves. In this way, the moving rod
functions as a cell. So this will be the source of
potential difference for the circuit.
Now, this question is asking us to
solve for 𝑣, the speed of the rod. So let’s rearrange this formula to
make 𝑣 the subject. To do this, we just need to divide
both sides by 𝑙𝐵 so those terms cancel out, leaving 𝑣 by itself on one side of
the equal sign. Thus, we have that 𝑣 equals the
induced emf divided by 𝑙𝐵.
We already know that 𝑙 equals 25
centimeters, or 0.25 meters, and that the strength of the magnetic field 𝐵 equals
945 milliteslas, or 0.945 teslas. We don’t know the magnitude of the
induced emf. So we need to find a way to express
it in terms of values that we do know. Notice that we were given the power
dissipated by the circuit, 𝑃, which equals 65.5 milliwatts, or 0.0655 watts. We can relate this power to emf
using the formula power equals the emf squared divided by 𝑅. And we can easily solve this
equation for the emf by multiplying both sides by 𝑅 and then taking the square root
of both sides. Thus, emf equals the square root of
𝑃 times 𝑅.
Now, we can substitute this
expression for the emf into the formula for speed. So it becomes 𝑣 equals the square
root of 𝑃𝑅 divided by 𝑙𝐵. We’re getting closer. We already know 𝑃, 𝑙, and 𝐵, but
we don’t yet know 𝑅, the total resistance of the circuit. So we’ll need to calculate it. Let’s refer to this value as 𝑅
total from now on.
To calculate the total resistance,
let’s note that the resistors 𝑅 one and 𝑅 two are connected in parallel to the
potential difference source and that the resistance of the rod can be modeled as a
resistor 𝑅 sub 𝑟 that’s connected in series with the potential difference
source. Thus, to find the total resistance
of the circuit, we’ll first find the equivalent resistance of 𝑅 one and 𝑅 two. So let’s recall the formula for
combining two resistors in parallel.
Substituting in those two
resistance values, we have that their equivalent resistance equals 11.52 ohms. Thus, the circuit’s total
resistance equals this value plus the resistance of the rod. We were told that the rod has a
resistance per unit length of 15 ohms per meter. And since the rod is 0.25 meters
long, we know that it has a resistance of 3.75 ohms. Therefore, the total resistance of
the circuit equals 11.52 ohms plus 3.75 ohms, or 15.27 ohms.
Finally, we’re ready to calculate
the speed of the rod 𝑣. Substituting in values for all the
terms on the right-hand side, we know that since they’re all expressed in their
appropriate base SI or SI-derived units, the units will work out nicely to give us a
speed expressed in meters per second. Now, grabbing a calculator, we get
a result of about 4.233 meters per second. And choosing to round this to one
decimal place, we have our final answer. We’ve found that the rod must be
moving through the magnetic field at a speed of 4.2 meters per second.
