In summary: So it is perpendicular to the edge of the circle. r is perpendicular to dl (since it points straight out of the yz plane), so it is also perpendicular to the circle.
  • #1
Sho Kano
372
3
When deriving the magnetic field strength due to a circular loop at some distance away from it's center (using Biot-Savart's law), why is the angle between ds and r 90 degrees?

This is a youtube video with the derivation, see 5:55
 
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  • #2
##\vec {dl}## is in the ##yz## plane. As drawn it is in the negative ##y## direction, so that ##\vec {dl} \perp \vec r## for the ##r## in the ##xz## plane. From Pythagoras you have ##R^2 + x^2 = r^2##. All ##\vec r## have the same length (from symmetry - rotation around the x-axis) - so this ##R^2 + x^2 = r^2## is true for all ##\vec r## -- so all these are rectangular triangles
 
  • #3
The 90 degrees comes from the cross product in Biot-Savart's law.

images?q=tbn:ANd9GcRON33tiq2s6fV0t1bSr3vau3hjKA34sHpOjW5YhE1z0ZBVNigw.png
 
  • #4
@Hesch: I would say the 90 degrees is used to simplify ##\vec a\times\vec b## to ##|\vec a||\vec b|## in BS, not that it comes from BS.
 
  • #5
BvU said:
##\vec {dl}## is in the ##yz## plane. As drawn it is in the negative ##y## direction, so that ##\vec {dl} \perp \vec r## for the ##r## in the ##xz## plane. From Pythagoras you have ##R^2 + x^2 = r^2##. All ##\vec r## have the same length (from symmetry - rotation around the x-axis) - so this ##R^2 + x^2 = r^2## is true for all ##\vec r## -- so all these are rectangular triangles
r is not pointing straight out of the yz plane, so why is it perpendicular?
 
  • #6
The ##\vec r## as drawn in the picture is in the xz plane. The ##\vec {dl}## as drawn is perpendicular to the xz plane: it is in the negative y direction. So that ##\vec {dl}## is perpendicular to that ##\vec r##.
For all other ##\vec {dl}## (in the yz plane) you can see they are ##\perp## 'their' ##\vec R## and also ##\perp## to the x-axis, so ##\perp## the plane in which their ##\vec R##, the x-axis and also ##\vec r##.

Bringing in Pythagoras was unnecessary - a mistake on my part.
 
  • #7
BvU said:
@Hesch: I would say the 90 degrees is used to simplify ⃗a×⃗b\vec a\times\vec b to |⃗a||⃗b||\vec a||\vec b| in BS, not that it comes from BS
Sorry, maybe it's a danish term.
Try to write something in danish to me. Maybe I will comment it. :wink:
 
  • #8
BvU said:
The ##\vec r## as drawn in the picture is in the xz plane. The ##\vec {dl}## as drawn is perpendicular to the xz plane: it is in the negative y direction. So that ##\vec {dl}## is perpendicular to that ##\vec r##.
For all other ##\vec {dl}## (in the yz plane) you can see they are ##\perp## 'their' ##\vec R## and also ##\perp## to the x-axis, so ##\perp## the plane in which their ##\vec R##, the x-axis and also ##\vec r##.

Bringing in Pythagoras was unnecessary - a mistake on my part.
So just because yz is perpendicular to xz? There is no way dl is perpendicular to r because r extends from the yz plane down to xy, making an angle that is not 90 degrees.
 
  • #9
No, not just because yz ##\perp## xz. Because ##\hat y \perp## xz for the situation as drwn in the video. In general:

Point P goes around the ring. At all such P $$ \vec {dl}\perp \vec R \quad \& \quad \hat x \perp \vec {dl} \quad \Rightarrow \quad \vec {dl}\perp (\vec x - \vec R) = \vec r $$

upload_2016-5-6_12-55-15.png


Hesch: jeg forstar lidt Dansk
 

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  • #10
BvU said:
No, not just because yz ##\perp## xz. Because ##\hat y \perp## xz for the situation as drwn in the video. In general:

Point P goes around the ring. At all such P $$ \vec {dl}\perp \vec R \quad \& \quad \hat x \perp \vec {dl} \quad \Rightarrow \quad \vec {dl}\perp (\vec x - \vec R) = \vec r $$

View attachment 100340

Hesch: jeg forstar lidt Dansk
I lost you at dl perpendicular to x - R, why did you do that? And why is it equal to r?
 
  • #11
dl is perpendicular to ##\vec R##
dl is perpendicular to ##\hat x## so dl is perpendicular to ##\vec x##
that means dl is perpendicular to the plane in which both ##\vec R## and ##\vec x## lie.
That plane is described by ##a\vec x + b \vec R##.
The vector ##\vec r## is ##\vec R -\vec x## so it is in that plane.
Why is ##\vec r=\vec x -\vec R## ?
i don't know how to say that. Perhaps a picture helps ?

upload_2016-5-6_22-46-10.png
 

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  • #12
BvU said:
dl is perpendicular to ##\vec R##
dl is perpendicular to ##\hat x## so dl is perpendicular to ##\vec x##
that means dl is perpendicular to the plane in which both ##\vec R## and ##\vec x## lie.
That plane is described by ##a\vec x + b \vec R##.
The vector ##\vec r## is ##\vec R -\vec x## so it is in that plane.
Why is ##\vec r=\vec x -\vec R## ?
i don't know how to say that. Perhaps a picture helps ?

View attachment 100362
Okay, I see where x - R comes from.

So this is how I'm comprehending it:
dl is pointing outwards (imagine a wire pointing out), so any r from its surface is perpendicular to it.
Before, I was confusing dl with the z vector. The angle that r makes with z will not be perpendicular.
 
  • #13
I think the reason is that they look for field only along the ##z## axis. I guess there the integral can be solved analytically, while the field of the current-conducting circular loop at an arbitrary position leads to elliptic integrals.
 

1. What is the formula for calculating magnetic field strength for a circular loop?

The formula for calculating magnetic field strength for a circular loop is B = μ0I / 2R, where B is the magnetic field strength, μ0 is the permeability of free space, I is the current flowing through the loop, and R is the radius of the loop.

2. How does the magnetic field strength vary with the radius of the circular loop?

The magnetic field strength varies inversely with the radius of the circular loop. This means that as the radius increases, the magnetic field strength decreases and vice versa.

3. Can the direction of the magnetic field strength be changed by changing the direction of the current in the circular loop?

Yes, the direction of the magnetic field strength can be changed by changing the direction of the current in the circular loop. The magnetic field lines will follow the right-hand rule, where the thumb points in the direction of the current and the fingers curl in the direction of the magnetic field.

4. How does the number of turns in a circular loop affect the magnetic field strength?

The magnetic field strength is directly proportional to the number of turns in a circular loop. This means that as the number of turns increases, the magnetic field strength also increases.

5. What is the unit of measurement for magnetic field strength?

The unit of measurement for magnetic field strength is Tesla (T). However, for smaller magnetic fields, the unit of Gauss (G) is often used. 1 T = 10,000 G.

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