Determining the B-field in center of current carrying loop

[kaashmonee]

Homework Statement

Determine the B-field inside the middle of a circular loop of current.

Homework Equations

Attempt at using Ampere's law: $\oint \vec{B} \cdot d \vec{l} = \mu_0 i$

The Attempt at a Solution

$\oint B \cdot R d \theta = \mu_0 i \Rightarrow BR(2 \pi) = \mu_0 i \Rightarrow B = \frac{\mu_0 i}{2 \pi R}$ How do I get the $2 \pi$ term to cancel out when doing this with Ampere's law?

 

[kuruman]

How do I get the $2 \pi$ term to cancel out when doing this with Ampere's law?

You don't. Although Ampere's law is always valid, the symmetry of the problem is not high enough to allow calculation of the B-field using it. You have to use the Law of Biot-Savart.

 

[Orodruin]

To be more specific, you have gone about using Ampere's law in a way different from what it is saying. Instead of looking at the B-field in a point and summing up the contributions from all parts of the loop, what you have done is to integrate the B-field along the loop and considered the current as a current enclosed by that loop. This procedure will give you the B-field at a distance $R$ from a straight long infinite wire carrying a current $i$, not the B-field in the center of a loop carrying the current $i$. I believe this should serve as a good warning that you cannot apply formulae blindly without understanding what they are saying.

As @kuruman said, try applying Biot-Savart's law instead and be careful in what it says to interpret the result correctly.

 

[kaashmonee]

Awesome, thank you for your replies! But I'm still wondering why it would be incorrect to take a tiny dl of the current carrying loop, draw an amperian loop around it, and add up the upward facing components of the B field?

 

[Orodruin]

Awesome, thank you for your replies! But I'm still wondering why it would be incorrect to take a tiny dl of the current carrying loop, draw an amperian loop around it, and add up the upward facing components of the B field?

Again, this is not what Ampere's law is telling you. Ampere's law is about the magnetic field at the Amperian loop and the current that passes through it. It is not about the current at the loop and the magnetic field inside it. What you need is Biot-Savart's law that tells you what the contribution to the magnetic field at a given point is from a small current carrying segment somewhere else. You indeed get the full magnetic field by summing those contributions, i.e., integrating.

 

[kuruman]

But I'm still wondering why it would be incorrect to take a tiny dl of the current carrying loop, draw an amperian loop around it, and add up the upward facing components of the B field?

Whatever Amperian loop you use, it has to enclose the current and it has to pass through the point where you want to find the B-field (center of the loop) and add the tangential component of the B-field around the complete loop, not just a "tiny" element dl. That's what $\oint \vec B \cdot d\vec l$ is saying you should do. Ampere's law is useful for finding the B-field when two conditions are met, (a) $\vec B$ is either parallel or perpendicular to $d \vec l$ in which case $\vec B \cdot d\vec l= B~dl$ or $\vec B \cdot d\vec l= 0$ and (b) $B$ is constant on the loop so that it can be taken out of the integral, in which case $\oint dl=C$ where $C$ is the perimeter of the loop. In this problem, if you consider an Amperian loop that encloses the current, neither of these conditions is met and that's why you cannot use Ampere's law. I repeat that the fact that you cannot use Ampere's law in this case does not invalidate Ampere's law. If you could do the line integral, say numerically, you would indeed get a number that is equal to $\mu_0i$.

Now to complete the picture for you, if you make the radius of the Amperian loop very very tiny so that it is much less than the radius of the wire loop, the circle locally will look like a long straight wire much like a straight line drawn parallel to the surface of the Earth looks "straight" at small distances above it. In this approximation, your calculation will give what you got in your original post. However, that is the field very near the circumference of the loop and very far from the center of the wire loop, which is the point where you want to find the B-field.