Magnetic field in a circular loop

[horsedeg]

Homework Statement

A conductor consists of a circular loop of radius R and two long, straight sections as shown in the figure. The wire lies in the plane of the page and carries a current I.

(a) What is the direction of the magnetic field at the center of the loop?
(b) Find an expression for the magnitude of the magnetic field at the center of the loop. (Use any variable or symbol stated above along with the following as necessary: μ0 and π.)

Homework Equations

Biot-Savart law
Ampere's Law?

The Attempt at a Solution

For (a), I'm not 100% sure if this reasoning is right, but using a right-hand rule, I pointed my thumb in the direction of the current, which is clockwise, then checked which way my fingers were curling, which would be kind of towards the center of the loop and thus into the page. It doesn't really make 100% sense to me, but it seems to produce the right answer.

For (b), I'm really confused about one thing. The solution says that the total magnetic field is the combination of the field due to the long straight wire and the field due to the circular loop. How are these any different at all? If you use the Biot-Savart law, you get a certain answer (μI/2R). But supposedly the answer is that plus the same answer with π on the bottom. So I guess I have the field from the circular loop. How do you get the other one?

 

[Simon Bridge]

(a) ... what is the trouble? If you do not follow what the "direction the fingers curl" approach works, you can always try the vector-calculus method directly.
(b) ... Look up "magnetic field of a long straight wire"... or just calculate it using the laws you listed.

 

[horsedeg]

Okay, so I think I'm missing something really basic.

So basically, at that point, we are adding the magnetic field due to the circle, AND due to the straight wire? I've been very confused about this because I thought there were two separate magnetic fields from the same wire. Assuming this is true, I understand this now.

 

[Simon Bridge]

There is only one magnetic field from the current in the wire.
You could, if you really want to, derive that magnetic field at the position indicated by applying the laws from scratch and doing the vector calculus.
However, in the process of doing that (give it a go!) you will discover that you can divide the mathematics out into an integral for the straight section added to another one for the loop section.
Since you already know the solutions for those, you can just plug them in.
This is called "the superposition principle"... and it saves a lot of work.
All it means is that the single B field for the wire and loop is the resulting field from the sum of of those for the loop and the wire separately.
Just like a difficult sum can be broken into easier steps.

 

[horsedeg]

You could, if you really want to, derive that magnetic field at the position indicated by applying the laws from scratch and doing the vector calculus.
However, in the process of doing that (give it a go!) you will discover that you can divide the mathematics out into an integral for the straight section added to another one for the loop section.

Yeah, I see what you are saying. There is already a formula to find the magnetic field of a long straight wire, so I just combine it with the circular field at that point and get the answer.

So here's the thing I don't get though.

Let's take this straight wire and say it has an infinite length. At that point, there is a magnetic field (I didn't draw the direction but you get the point). Why does the length of the wire matter at all? How is the field at that point affected any more than it is if it was just a tiny wire with an infinitesimal length dx but the same radius and current? It makes sense in a circle, but wouldn't it be unaffected without any changes in angle?

 

[Simon Bridge]

Let's take this straight wire and say it has an infinite length. At that point, there is a magnetic field (I didn't draw the direction but you get the point). Why does the length of the wire matter at all? How is the field at that point affected any more than it is if it was just a tiny wire with an infinitesimal length dx but the same radius and current?

... I hear you. The difference is in the limits of the integral in the biot savart law. Basically, the magnetic field $\vec B(r,x)$ for your example, depends on the current through the entire wire. You should be able to intuit why this has to be the case when you see what happens if the current is not always in the same direction.
So the length of the wire must be important ... the reason there is no position dependence for the infinite wire is because there is no central point on the wire and no ends to serve as special positions.

Also see discussion:
https://www.physicsforums.com/threads/magnetic-field-of-a-finite-length-wire.293994/

 

[horsedeg]

... I hear you. The difference is in the limits of the integral in the biot savart law. Basically, the magnetic field $\vec B(r,x)$ for your example, depends on the current through the entire wire. You should be able to intuit why this has to be the case when you see what happens if the current is not always in the same direction.
So the length of the wire must be important ... the reason there is no position dependence for the infinite wire is because there is no central point on the wire and no ends to serve as special positions.

Also see discussion:
https://www.physicsforums.com/threads/magnetic-field-of-a-finite-length-wire.293994/

I see. Looking at my notes now, my teacher proved this (I think) using Ampere's law. Is that a valid way? Here's how I did it:

But this doesn't seem to take into account the length.