# Magnetic Field of a current carrying loop

The numerical is given below:

Q: Compute the magnetic field of a long straight wire that has a circular loop with a radius of 0.05m. 2amp is the reading of the current flowing through this closed loop.

My teacher solved the above numerical as follows:

But I have a question that is:

Q: In ampere's law, there is a dot product of B and dl so where is cosθ? If Cosθ is not here, this means Cosθ = 1 which means that θ = 0° but how is θ = 0° when the B points outward or into the loop and dl is the small piece of the circumference of the loop?

If the solution is correct, please explain it and give the answer to my above query. If not correct, then please provide the correct solution to the problem.

EDIT:
There is no figure included in the numerical. We have to make ourselves. I think the figure of wire might be like this if the current loop itself is supposed as amperian loop:

• Please attach a figure of the wire. It is a bit confusing (atleast for me) as to what you mean by a straight wire with a circular loop. If it is semi-circular it would make sense. Kindly provide some clarifications regarding this. Commented Jul 16, 2020 at 8:18
• Theta is the angle between magnetic field, B, and the line normal to the plane that the loop of wire lies in. So I assume you’re meant to calculate the maximum value of the magnetic field which is when this angle theta is zero. Commented Jul 16, 2020 at 8:28
• @OliverGregory The angle theta is between the B and dl because Theta is the angle due to the dot product of B and dl. So how is theta the angle between magnetic field, B, and the line normal to the plane that the loop of wire lies in? Commented Jul 16, 2020 at 9:48
• @Lelouch I don't have a figure because this is a numerical assigned to me by my teacher in which there is no figure included. I have included figure which I have made. Commented Jul 16, 2020 at 9:58

The original question doesn't make sense with respect to the solution. The question should have been like,

Compute the magnitude of magnetic field due to a long straight wire at a distance of $$0.05\mathrm m$$ from it, given that the current in the wire is $$2A$$

You now see that the thing called as "loop" in the original is an imaginary loop known as amperian loop but not a wire.

Q: In ampere's law, there is a dot product of B and dl so where is cosθ? If Cosθ is not here, this means Cosθ = 1 which means that θ = 0° but how is θ = 0° when the B points outward or into the loop and dl is the small piece of the circumference of the loop?

You could see from the figure that indeed $$\cos\theta = 1$$, this video may be used for further information on ampere's law

• The confusion arises only because of this seemingly flawed statement - "a long straight wire that has a circular loop"
– user243016
Commented Jul 16, 2020 at 10:46
• Yes, if the question is like this then it's pretty clear that cosθ=1. Commented Jul 16, 2020 at 11:51
• just now i read that question completely and saw this phrase which adds even more confusion - "2amp is the reading of the current flowing through this closed loop"...giving respect to this phrase, if we consider a shape like unfolded paper clip(which has a so called "loop" on it), then too your solution process doesn't make sense because the magnetic field is different on different points in space and doesnt have an unique value.
– user243016
Commented Jul 16, 2020 at 12:07
• Hence the question must be wrong!
– user243016
Commented Jul 16, 2020 at 12:10

The way I read this question, a straight wire comes in from infinity, makes a circular loop, and then continues on out to infinity (in the same direction). Given this, there would be two contributions to the field at the center of the loop. One from the straight segments which they calculated using Ampere's law, and one from the loop, which cannot be calculated using Ampere's law.