### Video Transcript

A length of wire is formed into a solenoid with ๐ turns of wire per millimeter. The wire carries a constant current ๐ผ. As a result, a magnetic field of strength ๐ต can be measured at the center of the solenoid. Which of the following changes to the system would increase the magnetic field strength at the center of the solenoid, assuming everything else remains constant? (A) Decreasing ๐, the number of turns of wire per millimeter. (B) Increasing the length of the solenoid by adding turns of wire while keeping ๐ constant. (C) Decreasing the length of the solenoid by removing turns of wire while keeping ๐ constant. (D) Increasing ๐ผ, the current in the wire. (E) Decreasing ๐ผ, the current in the wire.

The question is asking us about a length of wire thatโs formed into a solenoid. We can recall that when wire is formed into a solenoid, that means that it is shaped something like this so that it consists of a series of loops or turns. Weโre told in the question that this particular solenoid has ๐ of these turns of wire per millimeter. That means that if the length of the solenoid in millimeters is equal to capital ๐ฟ and it has capital ๐ turns of wire in total, then dividing capital ๐, the total number of turns, by capital ๐ฟ, the total length of the solenoid, we get the number of turns of wire per millimeter, lowercase ๐. Weโre also told that thereโs a constant current of ๐ผ in this wire and that as a result of this current, we have a magnetic field of strength ๐ต at the center of the solenoid.

Letโs recall that the magnetic field strength ๐ต anywhere inside a solenoid is equal to a constant ๐ naught, the permeability of free space, multiplied by capital ๐, the total number of turns of the wire, multiplied by the current ๐ผ in the wire divided by ๐ฟ, the solenoidโs length. Now weโre not given any information in the question about these quantities capital ๐ and capital ๐ฟ. Instead, the question is telling us about lowercase ๐, the number of turns of wire per millimeter. Since weโve seen that the turns per millimeter lowercase ๐ is equal to the total number of turns capital ๐ divided by the length capital ๐ฟ, then in this equation for the magnetic field strength ๐ต, we can replace the capital ๐ divided by the capital ๐ฟ with the lowercase ๐.

We have then that the magnetic field strength ๐ต anywhere inside a solenoid is equal to this constant ๐ naught multiplied by the current ๐ผ through the wire multiplied by the number of turns of wire per millimeter lowercase ๐. Since this quantity ๐ naught, the permeability of free space, is constant, then that means that the magnetic field strength ๐ต is proportional to ๐ผ multiplied by lowercase ๐. In this question, weโre being asked which of five proposed changes to the system would cause the magnetic field strength at the center of the solenoid to increase. In order to answer this, we can make use of this equation which relates this magnetic field strength ๐ต to the current ๐ผ and the number of turns per millimeter lowercase ๐.

Answer option (A) claims that we can increase the magnetic field strength at the center of the solenoid by decreasing ๐, the number of turns of wire per millimeter. However, this equation weโve got here says that the magnetic field strength ๐ต is directly proportional to the number of turns of wire per millimeter lowercase ๐. That is, if lowercase ๐ increases, then ๐ต will increase. But if lowercase ๐ decreases, then ๐ต will decrease. So then if we decrease lowercase ๐ as suggested in answer option (A), then weโll actually end up decreasing the magnetic field strength at the center of the solenoid, which is the opposite of what we want to do. That means that we know that answer option (A) cannot be correct.

Letโs now move on to answer options (B) and (C), which are talking about the overall length of the solenoid. Now in both cases, weโre keeping the number of turns per millimeter, lowercase ๐, constant. Answer option (B) says that we should be adding more turns of wire in order to increase the length of the solenoid, while answer option (C) says we need to remove turns of wire in order to decrease the solenoidโs length. Since in both cases weโre told that weโre keeping lowercase ๐ constant, that means that weโre keeping the spacing between consecutive turns of wire the same. So in answer option (B), weโre increasing the length ๐ฟ of the solenoid. But in order to do this, weโre adding more turns of wire so that weโre increasing the total number of turns capital ๐ in proportion to the increase in length.

Likewise, in answer option (C), weโre removing turns of wire in order to decrease the length capital ๐ฟ, which means that capital ๐ decreases in proportion to this decrease in length. In both cases, capital ๐ divided by capital ๐ฟ, which is equal to lowercase ๐, the number of turns of wire per millimeter, remains constant. We know that the magnetic field strength ๐ต is directly proportional to lowercase ๐ for a constant current ๐ผ. That means that if lowercase ๐ remains constant, then the magnetic field strength ๐ต must remain constant too. That means that whether weโre increasing the length of the solenoid by adding turns of wire or weโre decreasing it by removing turns of wire, then so long as the number of turns per millimeter remains constant, then the magnetic field strength inside the solenoid wonโt change.

We can see then that the changes proposed in answer options (B) and (C) wonโt actually affect the magnetic field strength at the center of the solenoid at all. And so neither of these two answer options can be correct.

This leaves us with answer options (D) and (E), which are both talking about the current in the wire. Answer option (D) says that in order to increase the magnetic field strength at the center of the solenoid, we need to increase this current ๐ผ, while answer option (E) says that we need to decrease it. If we look again at this equation relating the field strength ๐ต, the current ๐ผ, and the turns per millimeter lowercase ๐, then we can see that if lowercase ๐ remains constant, ๐ต is directly proportional to ๐ผ. That means that if the current ๐ผ increases, then the magnetic field strength ๐ต must also increase, while if ๐ผ decreases, then ๐ต must also decrease.

So if we increase the current ๐ผ as suggested in answer option (D), then we will increase the magnetic field strength ๐ต at the center of the solenoid. Meanwhile if as suggested in answer option (E) we decrease this current ๐ผ, then weโll end up decreasing the magnetic field strength. That means that we know option (E) is not correct. We choose option (D) as our answer. The change to the system that would increase the magnetic field strength at the center of the solenoid is increasing ๐ผ, the current in the wire.