3

While reading Ampere's circuital law, I saw the equation for the field inside a toroid being given as $$B=\frac{\mu_0 NI}{2\pi r}$$ where

$ N=$ number of turns of toroidal coil

$I=$ current carried by the coil

$r=$ average radius of the coil

My question is how does this choice of average 'radius' affect the computation of the field inside the coil? Is there a formula to give the exact field at any arbitrary point inside the coil?

Thank you.

QFTheorist
  • 854
  • Essentially you are asking what if we relax the assumption that the radius of the coil cross-section is much smaller than the radius of the toroid? Note, that $r$ is half the diameter of the toroid - calling it the radius of the coil is a recipe for confusion. – ProfRob Feb 15 '20 at 08:52

1 Answers1

4

Actually the above is the formula for the exact field within the toroidal winding if we neglect things like finite wire diameter and circumferential current. You can see this because reflection through a plane containing the principal axis of symmetry of the toroid doesn't change the current distribution, so it doesn't change the magnetic field either. However such a mirror reflection reverses the in-plane components of $\vec B$ and leaves the out of plane components unchanged because $\vec B$ is an axial vector field. Thus only the out of plane, i.e. azimuthal component of $\vec B$ can be nonzero. Then rotational symmetry tells us that $B_{\phi}$ doesn't depend on the azimuthal angle $\phi$ so Ampere's law gets us exactly the $\vec B$ field.

To get flux through a loop of the coil we might use a formula like $\Phi_1=\frac{\mu_0NIA}{2\pi}\left(\frac1r\right)_{\text{average}}$ where $A$ is the cross-sectional area of the coil. For a toroid with circular cross section of radius $b$ centered a distance $a$ from the principal axis of symmetry, we have $$\Phi_1=\int_0^{2\pi}\int_0^b\frac{\mu_0NI}{2\pi(a-r\cos\theta)}r\,dr\,d\theta=\frac{\mu_0NIb^2}{a+\sqrt{a^2-b^2}}=\frac{\mu_0NI\pi b^2}{2\pi}\left(\frac1r\right)_{\text{average}}$$ So for this geometry we get $$\left(\frac1r\right)_{\text{average}}=\frac2{a+\sqrt{a^2-b^2}}$$ For a square cross section with side $2b$ centered a distance $a$ from the principal axis of symmetry we get $$\Phi_1=\int_{-b}^b\int_{-b}^b\frac{\mu_0NI}{2\pi(a-x)}dx\,dy=\frac{2b\mu_0NI}{2\pi}\ln\left(\frac{a+b}{a-b}\right)=\frac{4b^2\mu_0NI}{2\pi}\left(\frac1r\right)_{\text{average}}$$ So for this geometry $$\left(\frac1r\right)_{\text{average}}=\frac1{2b}\ln\left(\frac{a+b}{a-b}\right)$$ In either case the answer approaches $1/a$ as $b\rightarrow0$ but the exact form depends on coil geometry.

user5713492
  • 439