# Question Video: Electromagnetic Interactions between Straight Conductors Physics

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 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

04:43

### 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.