Video Transcript
The diagram shows concentric
field lines of the magnetic fields of two parallel current-carrying
conductors. Both currents are into the
plane of the diagram. And the currents have the same
magnitude as each other. The increase in radius of the
concentric field lines is constant. And the strength of a magnetic
field at a point around a current is proportional to the perpendicular distance
of the point from the current. Which of the following list of
the points shown in the diagram correctly orders the points from greatest to
smallest magnitude of net magnetic fields? (A) 𝐵, 𝐶, 𝐸, 𝐴, 𝐷. (B) 𝐵, 𝐸, 𝐶, 𝐴, 𝐷. (C) 𝐴, 𝐵, 𝐶, 𝐷, 𝐸. (D) 𝐸, 𝐵, 𝐶, 𝐴, 𝐷. (E) 𝐷, 𝐴, 𝐸, 𝐶, 𝐵.
On the diagram, we see the two
wires set up on the 𝑥-axis along with the five points from 𝐴 to 𝐸. The dotted circles around each
wire represent the magnetic fields. To determine which points are
greater and smaller in magnitude of net magnetic fields, we need to remember two
things. First, when we’re finding the
net magnetic field, we need to use vector addition. This means that if the magnetic
field lines are pointing in the same direction at the same position, we will add
the magnitudes together. And if they’re pointing in the
opposite direction of each other, we will subtract the magnitudes.
Let’s apply this to our diagram
by using the right-hand rule. We need to remember that we can
use the right-hand rule to find the direction of the magnetic field around a
current-carrying wire, where our thumb will point in the direction of the
current and our fingers will curl around the wire to show the direction of the
field lines. When we do the right-hand rule
on our diagram, our thumb points into the screen and our fingers curl
around. This shows that our magnetic
fields will be pointing clockwise around both wires. We can use yellow to show the
direction of the field around the right wire and pink to show the direction of
the field around the left wire.
Let’s look how both fields are
oriented at the five points on our 𝑥-axis. At position 𝐴 and 𝐵, both of
our fields are pointing towards the top of the screen. So the magnitude of the field
will be added together. At position 𝐶 and 𝐷, the
field lines are pointing in opposite directions. Yellow is pointing to the top
of the screen, and pink is pointing to the bottom of the screen. So we would subtract the
magnitudes. Both fields are pointing to the
bottom of the screen at point 𝐸, which means that we will once again add the
magnitudes together. Point 𝐷 is in the middle of
the wire. And since we know that the
magnitude of the magnetic fields is being subtracted at this position, we can
say that the net magnetic field at 𝐷 is zero.
With the net magnetic field at
point 𝐷 being the smallest, we can eliminate any answer choices that don’t have
the last letter as 𝐷. This would be answer choices
(C) and (E). To differentiate between any of
the other points, we need to remember the equation for the magnetic field around
a current-carrying wire. The magnetic field 𝐵 is equal
to the magnetic permeability, 𝜇 naught, times the current in the wire, 𝐼,
divided by two 𝜋𝑟, where 𝑟 is the distance between the position and the
wire. In this example, the current 𝐼
of both wires is the same.
Therefore, no matter where we
are in our diagram, we will have the same current, 𝐼, and our magnetic
permeability, 𝜇 naught, as a constant as well as two 𝜋. Therefore, the magnetic field
will be proportional to one over 𝑟. Looking at our answer choices,
we might be able to solve our problem if the net magnetic field is greater at 𝐸
than at 𝐵 by starting to compare those two points.
Let’s label the wires one and
two to differentiate them, with one being the left wire and two being the right
wire. 𝐸 is one radii, 𝑟, away from
the closest wire, in this case wire two, whereas 𝐵 is just shy of two radii
away from its closest wire, wire one. Comparing the farther wires, 𝐸
is approximately four radii away from its farthest wire, wire one, where 𝐵 is
just shy of five radii away from its farther wire, wire two. Point 𝐸 is closer to the
nearest wire as well as the farthest wire when compared to point 𝐵. And at both 𝐸 and 𝐵, we would
add the magnitude of the two magnetic fields together because they’re pointing
in the same direction.
If we are adding the magnitudes
together and 𝐸 is closer, meaning a smaller 𝑟, there will be a bigger net
magnetic field at 𝐸 than there would be at 𝐵. With the net magnetic field at
point 𝐸 being larger than the net magnetic field at point 𝐵, our final answer
will be choice (D), that the magnitude from greatest to smallest of the net
magnetic field is 𝐸, 𝐵, 𝐶, 𝐴, 𝐷.
