Question Video: Understanding the Relationship between Current and Magnetic Field for a Solenoid | Nagwa Question Video: Understanding the Relationship between Current and Magnetic Field for a Solenoid | Nagwa

# Question Video: Understanding the Relationship between Current and Magnetic Field for a Solenoid Physics • Third Year of Secondary School

## Join Nagwa Classes

A length of wire is formed into a solenoid with ๐ turns of wire per millimeter. The wire carries a constant current ๐ผ. As a result, a magnetic field of strength ๐ต 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 ๐ผ, the current in the wire [E] Decreasing ๐ผ, the current in the wire

06:24

### Video Transcript

A length of wire is formed into a solenoid with ๐ turns of wire per millimeter. The wire carries a constant current ๐ผ. As a result, a magnetic field of strength ๐ต 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 ๐ผ, the current in the wire. (E) Decreasing ๐ผ, the current in the wire.

The question is asking us about a length of wire thatโs 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. Weโre 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 ๐ฟ and it has capital ๐ turns of wire in total, then dividing capital ๐, the total number of turns, by capital ๐ฟ, the total length of the solenoid, we get the number of turns of wire per millimeter, lowercase ๐. Weโre also told that thereโs a constant current of ๐ผ in this wire and that as a result of this current, we have a magnetic field of strength ๐ต at the center of the solenoid.

Letโs recall that the magnetic field strength ๐ต 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 ๐ผ in the wire divided by ๐ฟ, the solenoidโs length. Now weโre not given any information in the question about these quantities capital ๐ and capital ๐ฟ. Instead, the question is telling us about lowercase ๐, the number of turns of wire per millimeter. Since weโve seen that the turns per millimeter lowercase ๐ is equal to the total number of turns capital ๐ divided by the length capital ๐ฟ, then in this equation for the magnetic field strength ๐ต, we can replace the capital ๐ divided by the capital ๐ฟ with the lowercase ๐.

We have then that the magnetic field strength ๐ต anywhere inside a solenoid is equal to this constant ๐ naught multiplied by the current ๐ผ 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 ๐ต is proportional to ๐ผ multiplied by lowercase ๐. In this question, weโre 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 ๐ต to the current ๐ผ 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 weโve got here says that the magnetic field strength ๐ต is directly proportional to the number of turns of wire per millimeter lowercase ๐. That is, if lowercase ๐ increases, then ๐ต will increase. But if lowercase ๐ decreases, then ๐ต will decrease. So then if we decrease lowercase ๐ as suggested in answer option (A), then weโll 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.

Letโs now move on to answer options (B) and (C), which are talking about the overall length of the solenoid. Now in both cases, weโre 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 solenoidโs length. Since in both cases weโre told that weโre keeping lowercase ๐ constant, that means that weโre keeping the spacing between consecutive turns of wire the same. So in answer option (B), weโre increasing the length ๐ฟ of the solenoid. But in order to do this, weโre adding more turns of wire so that weโre increasing the total number of turns capital ๐ in proportion to the increase in length.

Likewise, in answer option (C), weโre removing turns of wire in order to decrease the length capital ๐ฟ, which means that capital ๐ decreases in proportion to this decrease in length. In both cases, capital ๐ divided by capital ๐ฟ, which is equal to lowercase ๐, the number of turns of wire per millimeter, remains constant. We know that the magnetic field strength ๐ต is directly proportional to lowercase ๐ for a constant current ๐ผ. That means that if lowercase ๐ remains constant, then the magnetic field strength ๐ต must remain constant too. That means that whether weโre increasing the length of the solenoid by adding turns of wire or weโre 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 wonโt change.

We can see then that the changes proposed in answer options (B) and (C) wonโt 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 ๐ผ, while answer option (E) says that we need to decrease it. If we look again at this equation relating the field strength ๐ต, the current ๐ผ, and the turns per millimeter lowercase ๐, then we can see that if lowercase ๐ remains constant, ๐ต is directly proportional to ๐ผ. That means that if the current ๐ผ increases, then the magnetic field strength ๐ต must also increase, while if ๐ผ decreases, then ๐ต must also decrease.

So if we increase the current ๐ผ as suggested in answer option (D), then we will increase the magnetic field strength ๐ต at the center of the solenoid. Meanwhile if as suggested in answer option (E) we decrease this current ๐ผ, then weโll 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 ๐ผ, the current in the wire.

## Join Nagwa Classes

Attend live sessions on Nagwa Classes to boost your learning with guidance and advice from an expert teacher!

• Interactive Sessions
• Chat & Messaging
• Realistic Exam Questions