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Question Video: Finding the Number of Turns of Wire per Unit Length of a Solenoid Physics

A solenoid is formed of a wire that carries a constant current of 0.16 A. The magnetic field at the center of the solenoid is measured to be 3.8 × 10⁻⁴ T. Calculate the number of turns of wire per centimeter of the solenoid’s length, rounding to the nearest whole number of turns. Use a value of 4𝜋 × 10⁻⁷ T⋅m/A for 𝜇₀.

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Video Transcript

A solenoid is formed of a wire that carries a constant current of 0.16 amperes. The magnetic field at the center of the solenoid is measured to be 3.8 times 10 to the negative four tesla. Calculate the number of turns of wire per centimeter of the solenoid’s length, rounding to the nearest whole number of turns. Use a value of four 𝜋 times 10 to the negative seven tesla meters per ampere for 𝜇 naught.

This question is asking us about a solenoid, which is a wire that’s formed into a series of equally spaced loops or turns as illustrated here. We’re told that the wire that forms this solenoid carries a constant current of 0.16 amperes, which we’ve labeled as 𝐼. As a result of this current, there’s a magnetic field inside of the solenoid. We’re told that the strength of this magnetic field, which we’ve labeled as 𝐵, is measured at the center of the solenoid to be equal to 3.8 times 10 to the negative four tesla. The magnetic field strength 𝐵 inside of a solenoid depends on the current 𝐼 through the wire, but it also depends on the solenoid’s total length, capital 𝐿, and capital 𝑁, the total number of turns of wire forming the solenoid.

More precisely, the magnetic field strength 𝐵 is equal to a constant 𝜇 naught, the permeability of free space, multiplied by capital 𝑁, the total number of turns of wire, multiplied by the current 𝐼 divided by the solenoid’s length, capital 𝐿. Now, in this equation, we know the value of the magnetic field strength 𝐵, and we’re also given the current 𝐼 in the wire. 𝜇 naught is a constant that we’re given the value of. This leaves us with the number of turns, capital 𝑁, and the length of the solenoid, capital 𝐿. We’re not given values for either of these two quantities, and neither of them is the quantity that we are asked to find in the question.

We are asked for the number of turns of wire per centimeter of the solenoid’s length. Let’s label the number of turns of wire per unit length of the solenoid as lowercase 𝑛. We can see that the number of turns per unit length must be equal to the total number of turns, capital 𝑁, divided by the total length, capital 𝐿. We can then use this relationship in order to replace the capital 𝑁 divided by capital 𝐿 in this equation by lowercase 𝑛. Making this substitution, we have that 𝐵 is equal to 𝜇 naught multiplied by lowercase 𝑛 multiplied by 𝐼. Now, we know the magnetic field strength 𝐵 and the current 𝐼, and we’ve got a value for the permeability of free space, 𝜇 naught.

We want to solve this equation for lowercase 𝑛, the number of turns of wire per unit length. To make lowercase 𝑛 the subject of the equation, we divide both sides by 𝜇 naught and 𝐼. Then, on the right-hand side, the 𝜇 naughts and 𝐼s cancel from the numerator and the denominator. Swapping over the left- and right-hand sides of the equation, we can then rewrite it as lowercase 𝑛 is equal to 𝐵 divided by 𝜇 naught 𝐼. We can now take our values that we’re given for the magnetic field strength 𝐵 and the current 𝐼 along with the value we’re given for 𝜇 naught and substitute them into this equation. When we do that, we get this expression here for lowercase 𝑛, the number of turns per unit length. That’s 3.8 times 10 to the negative four tesla divided by four 𝜋 times 10 to the negative seven tesla meters per ampere and 0.16 amperes.

If we now take a look at the units of the quantities on the right-hand side, we can see that in the denominator of the fraction, the amperes and the per ampere cancel each other out. Then, the units of tesla in the numerator cancel with the units of tesla in the denominator. That means that the only units left are the units of meters in the denominator of the fraction. That means that using these values, we’ll calculate a value for lowercase 𝑛 with overall units of meters to the negative one or units of per meter. That is, the value for lowercase 𝑛 that we’ll find will be the number of turns of wire per meter of the solenoid’s length. Evaluating the expression then gives a result for lowercase 𝑛 of 1.88996 et cetera times 10 to the three per meter.

We should notice though that we weren’t asked for the number of turns of wire per meter, but rather the number of turns of wire per centimeter. That is, we need to convert this value that we found for lowercase 𝑛 from units of meters to the negative one into units of centimeters to the negative one. In order to do this, we can start by recalling that one meter is equal to 100 centimeters. If we divide both sides of this relationship by one meter and by 100 centimeters so that the one-meter terms on the left cancel out and the 100-centimeter terms on the right cancel out, we have that one over 100 centimeters is equal to one over one meter.

Since one over units of centimeters is centimeters to the negative one and one over units of meters is meters to the negative one, then we have that one over 100 centimeters to the negative one is equal to one meter to the negative one. What this equation here means is that in order to convert a quantity from units of meters to the negative one into units of centimeters to the negative one, we need to divide by a factor of 100.

Let’s now clear ourselves some more space so that we can use this to calculate a value for lowercase 𝑛 in units of centimeters to the negative one. We know that lowercase 𝑛 is equal to 1.88996 et cetera times 10 to the three meters to the negative one. And we also know that to convert from meters to the negative one to centimeters to the negative one, we divide by 100. Dividing by a factor of 100 and evaluating the expression gives us that lowercase 𝑛 is equal to 18.8996 et cetera centimeters to the negative one. That is, the number of turns of wire per centimeter of this solenoid is equal to 18.8996 and so on.

The last step left to go is to notice that we are asked for our answer to the nearest whole number of turns. Rounding to the nearest whole number gives us our answer that there are 19 turns of wire per centimeter of the solenoid’s length.

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