# Do Ampère's and the the Biot-Savart law give different answers for the magnetic field of a circular loop? [duplicate]

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I have been trying to calculate the magnetic field due to a circular loop using Ampère's circuital law. However, I am unable to do the same.

Whenever I apply the law, while integrating length element $\mathrm{d}l$, I get $2\pi r$ [the circumference of the loop]. However, when we apply Biot - Savart Law, the $\pi$ from the circumference gets cancelled with the $\pi$ from the constant of proportionality. Please do point out the error in my derivation.

I would also like to know if the usage of Biot-Savart law is any different from Ampere's Law when considering the restrictions or required conditions.

## marked as duplicate by ACuriousMind♦, Kyle Kanos, John Rennie, Qmechanic♦Apr 10 '15 at 15:13

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## 2 Answers

Ampere's law is not useful in this case. It says that the line integral of the B field around a closed path is equal to $\mu_0$ times the current passing through the closed path (for steady currents).

To use the law you want the LHS to be simple to evaluate. Usually the B field is constant in magnitude around the path and either parallel or perpendicular to the path. In this case you cannot arrange this. As you can easily show with the Biot-Savart law, the B field varies with distance from the centre of the current loop, so it is difficult to define a simple line integral path that encloses the current.

• I agree with you. So, I thought about incorporating and integrating with respect to theta (the angle between length element dl and radius r). So, I replaced the term dl with r times d(theta) and integrated the equation with the limit, 0 to 360 degrees (in raadians). That way, I could cancel the pi term and got a result similar to that obtained using the Biot-Savart law. However, I am not sure if this is the correct method. The assumption maid was that the value of theta was so small that cos( d(theta)) = d(theta). – phynerd Feb 4 '15 at 10:59
• @phynerd I'm not sure what loop you have chosen. If it is a circle that is concentric with the loop then B is perpendicular to the path and the line integral is zero (consistent with the fact that no current passes through it). If it is a circle around the wire, you have not accounted for the fact that B is a function of theta and at some theta-dependent angle to the path. You are not going to invent a new way to make this work - it doesn't. – Rob Jeffries Feb 8 '15 at 20:18

It is not at all obvious how Ampère's law would be used to to calculate the field at the centre of a circular loop of wire. Ampère's law tells that if we draw any closed curve in space then the magnetic field integrated along that curve is proportional to the current flowing through the space enclosed by the curve. Typically this is useful where we can choose a curve such that the magnetic field is constant along its length, because then the result of the integration is just $\ell B$, where $\ell$ is the length of the curve. I can't see any way to use this technique to calculate the field due to a circular loop of wire.

Your Biot-Savart result shouldn't contain $\pi$. You should get:

$$B = \frac{\mu_0 I}{2r}$$

• Than you for taking time to answer the question. My biot savart law result didn't contain the term pi. – phynerd Feb 4 '15 at 11:03