Question 7.6: A toroid whose dimensions are shown in Figure 7.15 has N tur...

We apply Ampère’s circuit law to the Amperian path, which is a circle of radius \rho shown dashed in Figure 7.15. Since N wires cut through this path each carrying current I, the net current enclosed by the Amperian path is NI. Hence,

\oint H\cdot dl=I_{enc}\rightarrow H\cdot 2\pi\rho=NI

or

H=\frac{NI}{2\pi\rho},   for  \rho_{o}-a\lt \rho\lt \rho_{o}+a

where \rho_{o} is the mean radius of the toroid as shown in Figure 7.15. An approximate value of H is

H_{approx}=\frac{NI}{2\pi\rho_{o}}=\frac{NI}{\ell}

Notice that this is the same as the formula obtained for H for points well inside a very long solenoid (\ell\gg a). Thus a straight solenoid may be regarded as a special toroidal coil for which \rho_{o}\rightarrow\infty . Outside the toroid, the current enclosed by an Amperian path is NI-NI=0 and hence H=0.