Solenoids and Toroids — Question 4

Question 4

Nested opposite solenoids can cancel a core field while leaving an annular field

Two ideal infinitely long coaxial solenoids have radii a and b, with b>a. Their turn densities are n1 and n2. The inner solenoid current I1 produces field along +z; the outer current I2 produces field along -z. Use vacuum permeability $\mu_0\simeq4\pi\times10^{-7}\,\mathrm{T\,m/A}$ where numerical values are needed.

1. Find B in the core, the annulus and the exterior.

2. For n1=1500 per metre, n2=1000 per metre and I1=.400 A, choose I2 to cancel the core field, then find the annular field.

Question 4 - Solution

Nested opposite solenoids can cancel a core field while leaving an annular field

(a) Method and calculation

$$r<a:\ \vec B=\mu_0(n_1I_1-n_2I_2)\hat z;\quad a<r<b:\ \vec B=-\mu_0n_2I_2\hat z.$$$$\boxed{\vec B=0\quad(r>b)}\quad\text{for ideal infinite windings}.$$

(b) Interpretation and further calculation

$$n_1I_1=n_2I_2\Rightarrow\boxed{I_2=.600\,\mathrm A}.$$$$\boxed{\vec B_{\mathrm{annulus}}=-(4\pi\times10^{-7})(600)\hat z=-.753982\,\mathrm{mT}\,\hat z}.$$

Physical check

Canceling the field in one spatial region does not cancel every region because each winding has a different field support in the ideal model.