1
$\begingroup$

I am referring to this problem in Griffiths:

enter image description here

enter image description here

My question: where does cos(theta) come from? r and dl are perpendicular to each other so with the formal from Biot-Savart, dlxr (vectors) = dl and nothing more. I see the explanation in parentheses but it still doesn't make sence.

Any insight is appreciated.

$\endgroup$
  • $\begingroup$ It is because you need the projection of $dB$ onto the $z$ axis. $\endgroup$ – Farcher Jun 13 at 21:07
  • $\begingroup$ $d\vec{l}\times\vec{r}$ is a vector, which needs to be split into components. $\endgroup$ – orion Jun 14 at 9:09
0
$\begingroup$

Look at a single dB. Now compare it to a dB contributed by an element of current on the opposite side of the ring. You will see their components that lie on the xy plane point opposite directions and thus cancel each other leaving you with just the z component of each dB. Going around the full circle does this to each "dB" contributed by a dl length of current. Cosine of theta will give you the adjacent side of the dB right triangle in the diagram which points along the z axis - this is the z component of dB. Theta is the same for all points along the ring so cosine of theta is a constant that can be moved outside the integral.

Also dl cross r does not equal dl. It equals dl multiplied by r (where the sine of the angle between dl and r is 90 degrees and so sine of 90 degrees is just 1). r is the same for all points on the ring (all dl's are the same distance away from the point in question) so r can be taken outside the integral and you are left with just integrating dl around the circle which is just the circumference of the circle

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.