Confusion regarding line integrals

In summary, the conversation discusses the difference in line integrals for a current-carrying wire and an Amperian loop in a uniform magnetic field. The net force on the current loop is zero due to the sum of elements pointing in different directions, while the Amperian loop is chosen so that the B-field is tangent to the loop, resulting in a non-zero value for the line integral. The speaker also mentions that the variable of integration would be from 0 to 2π as both integrals are taken over closed loops.
  • #1
rahularvind06
10
0
Sorry if this is the wrong place to post this, but I wasn't sure where exactly to put it.
When we calculate the force on a closed loop of current-carrying wire in a uniform magnetic field,
We calculate the line integral of the loop to be 0.
However, when we evaluate the line integral for an Amperian loop while using Ampere's law, we integrate from 0 to 2π.
Why is there a difference? I'm not very advanced with calculus, so some intuition is all I request.
 
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  • #2
I am not sure I understand what you are asking. Both line integrals are taken over closed loops, so if your variable of integration is some angle θ, then the limits of integration would be from 0 to 2π. The fact that the net force on a current loop in a uniform magnetic filed is zero is a different issue. Think of the integral as a sum. In the current loop case you are adding a whole bunch of elements ##d \vec{F}## that point in all sorts of directions so that their sum is zero regardless of the shape of the loop. In the Amperian loop case, you choose the loop so that the B-field is tangent to the loop in which case all the ##\vec{B}\cdot d\vec{\mathcal l}## terms add together to give a non-zero value.

I hope I answered your question.
 

1. What is a line integral?

A line integral is a type of integral used in multivariable calculus to calculate the total value of a scalar or vector field along a given curve or path.

2. What is the difference between a line integral and a regular integral?

A regular integral calculates the area under a curve in two dimensions, whereas a line integral calculates the value of a function along a path in three dimensions.

3. How do you calculate a line integral?

A line integral can be calculated by first parameterizing the curve or path and then integrating the function along that parameterization.

4. What is the significance of line integrals in science?

Line integrals have many applications in science, including calculating work done by a force, finding the mass of a wire or rope, and determining the circulation of a fluid flow.

5. Can line integrals be used to solve real-world problems?

Yes, line integrals are used in many real-world applications, such as calculating the amount of fluid flowing through a pipe or determining the electric field around a charged wire.

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