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Why is the magnetic field dot producted in the integral version of Ampere's circuital law?

Physics Asked on January 12, 2021

you know amperes circuital law? Well in that equation there’s a dot product between the magnetic field B and a length element dl…why is that? I mean its not like the magnetic field can be at an angle to the length element. Its always parallel to the length element so $cos(0)=1 $ which means the dot term is not needed in the equation

But yet it is there so I feel like I’m missing something fundamental or basic here…help would be great

2 Answers

I mean its not like the magnetic field can be at an angle to the length element. Its always parallel to the length element

This is not correct. The magnetic field certainly can be at an angle to the tangent line.

Consider a long straight wire carrying a steady current. If we take a circular path centered on the wire then, as you say, the field is always tangent to the path. Any other path will not have that property. Consider a square path around the wire, or a circular path outside the wire, or any other path. All of those will have places where the field is not tangent to the path.

The dot product is there to handle such paths.

Answered by Dale on January 12, 2021

Historical Note: It was because back then scientists humored the idea that a 'magnetic charge' be taken along a path and we seek its total 'work done', similar to how we do it in Electrostatics. Although our thoughts are now more refined than back then, the theory is still meaningful and its manifestations are all the more useful.

Answered by F.N. on January 12, 2021

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