Video Transcript
Several horizontal pairs of
parallel conducting wires are stacked vertically. The magnitude of the current in
each wire is the same. A cross section of the resultant
magnetic field due to the currents is shown in the diagram. Which of the configurations of
current direction shown would produce the resultant magnetic field? (A) I, (B) III, (C) I and III, (D)
I and IV, (E) III and IV.
In our diagram, we see this
arrangement of vertically stacked pairs of conducting wires. This is one pair here, and that
pair is stacked on top of this pair, which is stacked on top of this pair, and so
on. Each one of the wires in these
pairs carries current either into or out of the screen. Those directions are not indicated
in the diagram. What we do see though is a cross
section of the overall magnetic field formed by all these current-carrying
wires. Specifically, we see magnetic field
lines of this field.
Based on the appearance of these
field lines, we want to be able to choose which of our four answer options shows us
current direction that would generate a magnetic field that looks like this. To begin seeing how this can work,
let’s clear some space at the top of our screen and remind ourselves about the
magnetic fields that are produced by current-carrying wires.
Here, we imagine a wire carrying
current into the screen. That’s one possibility for the
wires in our diagram. And here, we have a wire carrying
current out of the screen. Generally speaking, given a
straight wire carrying a current 𝐼, then if we point the thumb on our right hand in
the direction of that current, the fingers of that hand will be able to curl closed
in the direction of the magnetic field around the wire. For this wire then, the magnetic
field would point in these directions.
Applying the same right-hand rule
to this wire carrying current into the screen, if we point the thumb on our right
hand in that same direction, then the fingers on our right hand can curl closed in
this direction, clockwise around the wire. This then is the magnetic field
direction around such a current. If we then consider the magnetic
field generated by a wire carrying current out of the screen towards us, pointing
our right thumb in that direction allows our fingers to curl closed this way,
counterclockwise.
Even though for each one of these
currents we’ve only drawn one closed field line, it’s worth noting that the magnetic
field from either one of these currents extends out infinitely far away from the
axis of the wire. This tells us that the magnetic
field, say, here right between the two wires is a combination of the magnetic field
generated by this wire and the magnetic field generated by this one.
Since the wire on the left
generates a clockwise-pointing magnetic field, at the location we’ve picked that
field will point straight downward. For the other current, the one that
points counterclockwise around the wire, note that at this location we’ve picked
that field will also point in the same direction straight down.
Magnetic field is a vector
quantity; it has magnitude and direction. If we add together these two
individual magnetic field vectors, then we would get a vector that looks something
like this. This represents the overall or the
net magnetic field midway between these wires.
We’re seeing then that given this
pair of current-carrying wires, the magnetic field between them will not be zero,
but rather it will be quite strong. On our diagram, areas of relatively
greater magnetic field strength are indicated by greater density of magnetic field
lines. Where there are lots of field lines
packed closely together, the field is relatively strong.
As a next step, let’s try stacking
an identical pair of current-carrying wires right below this one. Given another pair of wires like
this, the magnetic field lines around each one individually will point the same way
as before. And so if we consider the magnetic
field that’s formed between this particular pair of current-carrying wires, just
like for the pair above the net magnetic field will point strongly downward.
We can see then that as we stack
pairs of current-carrying wires like this one on top of another, what we get is a
magnetic field in between them with compounding strength. That is, the magnetic fields formed
by each individual pair add to the magnetic fields formed by the other pairs. If our diagram consisted of
current-carrying wires like these pairs stacked one on top of another, then we would
expect a large number of long, straight magnetic field lines running right through
the centers of the pairs. But what we see instead is that the
magnetic field lines seem to form loops around each current-carrying wire
individually.
We can say then that the current
directions in our configuration do not look like this pattern repeated over and
over. Notice that this is the current
direction configuration shown in answer option III. Because the field lines we see in
our diagram do not match the field lines in between the pairs of current-carrying
wires we see for the configuration demonstrated in answer option III, we can
eliminate this choice from consideration.
To test out a different
configuration of current-carrying wires, let’s say that we reverse the directions of
the current in each of the wires in this pair. That is, we’ll make the current in
the wire on the left come out of the screen. Our right-hand rule shows us that
would create a magnetic field like this. And we’ll let the current in the
other wire point now into the screen. This would yield a magnetic field
pointing clockwise.
Considering now the magnetic fields
only from this pair of wires, let’s now see what the net magnetic field is at a
point midway between them. The wire on the left, with a
counterclockwise magnetic field, would create a magnetic field at this point
pointing upward. And likewise, the magnetic field
from the current on the right at this location points in the same direction. If we add these magnetic fields
together, notice that we get a relatively stronger magnetic field pointing
upward. What we’re seeing is that the
magnetic field created by this pair of wires competes with the magnetic field
created by this pair above it, the overall effect being that the magnetic fields
caused by these stacked pairs of wires would more or less separate out along this
dashed line.
If we look back at our diagram, we
can see this happening there. The magnetic fields created by
these pairs of stacked wires are effectively compartmentalized to the region of that
pair. This tells us that if we were to
continue this alternating pattern of current-carrying pairs of wires — that is, if
the next pair in the stack carried currents like this and then the pair after that
like this — the overall effect would be to create a magnetic field that is
demonstrated in our diagram, that is, a field that is compartmentalized according to
each pair.
If we look for this pattern of
current-carrying wires among our answer options, we see that there’s a match with
answer choice I. Therefore, whatever letter we end
up choosing to represent our final answer must include this answer choice. This means we can eliminate option
(B), which doesn’t include I as well as option (E), which also doesn’t include this
configuration. Along with this, we can eliminate
answer choice (C). We could have done this earlier
because this choice includes answer option III.
The remaining answers raise the
question of whether configuration IV over here could also create the magnetic field
we see in our diagram. An important thing to notice about
this field is that we don’t know its direction at any point. In our sketches, we’ve drawn
arrowheads to indicate direction, but in our diagram, we don’t see any directional
indication. Given a closed loop of a magnetic
field line, say, this one right here, we don’t know then whether this loop points
clockwise or counterclockwise. This means that for all of our
current-carrying pairs, if we were to swap the left-side current direction with the
right-side current direction — that is, if the current directions would look like
this — then we would once again get an overall magnetic field that was effectively
segmented or compartmentalized by each pair.
The directions of the field lines
in each case will reverse, but recall that those directions aren’t indicated
here. And the pattern overall of the
field lines would remain the same. All this is to say that answer
choice IV is a legitimate configuration of current directions. Whether the current directions in
our wires look like this or like this, we would still get a magnetic field that
looks like the one we see in our diagram. Our final answer then we’ll be
option (D).
Before finishing up though, let’s
consider why configuration II is not a valid choice. If all of the wires in our
configuration carried current pointing out of the screen, then the magnetic field
around each one would point counterclockwise. If we consider then the magnetic
field formed midway between one of these pairs, we see that due to the current on
the left there is an upward pointing magnetic field, while due to the current on the
right there is an equal and oppositely directed magnetic field. These fields will combine to give a
net magnetic field of effectively zero. And the same thing will happen for
every other pair in our stack.
If our diagram matched this
configuration, we would expect to see almost no magnetic field lines in between the
pairs of wires. Because we do though, we can know
that configuration II is not a representation of the currents in these wires. Rather, it’s configurations I and
IV only that demonstrate current directions that would generate the magnetic field
shown.
