To see why an MRI utilizes iron to increase the magnetic field created by a coil, calculate the current needed in a 400-loop-per-meter circular coil 0.660 m in radius to create a 1.20-T field (typical of an MRI instrument) at its center with no iron present. The magnetic field of a proton is approximately like that of a circular current loop 0.650×1015 m0.650 \times 10^{-15} \textrm{ m} in radius carrying 1.05×104 A1.05 \times 10^4 \textrm{ A}. What is the field at the center of such a loop?
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Final Answer

2390 A2390 \textrm{ A}

1.01×1013 T1.01 \times 10^{13}  \textrm{T}

Solution video

OpenStax College Physics, Chapter 22, Problem 61 (Problems & Exercises)

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Video Transcript
This is College Physics Answers with Shaun Dychko. We're going to find the current that's needed in an MRI machine to create a magnetic field of 1.2 Tesla but assuming that there is no iron core and that we just have free space. So we use permeability of free space instead of the permeability of some you know, ferro-magnetic material which would increase the permeability. So we divide both sides of this formula for the magnetic field inside of a solenoid by mu naught and then the number of turns per meter, n. Then we have I equals B over mu naught n. So that's 1.2 Tesla divided by four pi times ten to the minus seven Tesla meters per amp, times four hundred loops per meter. This gives 2390 amps which is a massively high current and so that's why they like to use an iron core inside the solenoid of an MRI machine in order to increase this permeability and thereby decrease the current. They also use super-conducting wires so that, you know -- because even if you increase this number you're still going to have a large current and that reduces the power loss and the amount of heat generated by having super-conducting wires. This question also asks us to find out what magnetic field is produced by the spin of a proton. So the spin can be modeled as a circular loop with this current and this radius, and so we plug those numbers into our mu naught I over two r formula for the magnetic field due to the current carrying loop. So that's permeability of free space times 1.05 times ten to the four amps, divided by two times 0.65 femto-meters which is times ten to the minus fifteen meters. This gives 1.01 times ten to the thirteen Tesla.


wrong, there are multiple examples where you divide instead of multiply resulting in incorrect answers

Hi Jodaniel, thank you for your comment. This is probably related to your comments on problem 65 and Chap. 23, problem 11 as well. If I understand correctly you're inspecting the calculator screenshots, expecting to see parentheses surrounding the denominator and multiply signs between the factors in the denominator. That totally makes sense, and many students/teachers do it that way. To be clear, you're expecting to see, for example, 10/(2*5) = 1. This can also be written as 10/2/5 = 1. They both give the same answer since your calculator evaluates left to right: it calculates 10/2 first, gets the answer 5, then proceeds further to divide by 5, resulting in the correct answer 1. This left-to-right pattern applies only to operators with equal precedence (divide and multiply have equal precedence... I'm talking about order of operations "BEDMAS" here). I prefer the 10/2/5 pattern of button pushing since it requires fewer key presses.
Hope this helps!