Solenoids and Toroids — Question 10

Question 10

A gap between two semi-infinite solenoids reduces their axial field

Two thin coaxial solenoids of radius R and turn density n extend from $z=-\infty$ to $-g/2$ and from $+g/2$ to $+\infty$. Their currents I both produce +z fields. For a turn at axial distance u, use $dB=\mu_0nIR^2du/[2(R^2+u^2)^{3/2}]$. Use vacuum permeability $\mu_0\simeq4\pi\times10^{-7}\,\mathrm{T\,m/A}$ where numerical values are needed.

1. Integrate to find the field at the midpoint of the empty gap.

2. Find the gap g that reduces this field to half the uninterrupted infinite-solenoid value.

Question 10 - Solution

A gap between two semi-infinite solenoids reduces their axial field

(a) Method and calculation

$$B(0)=2\int_{g/2}^{\infty}\frac{\mu_0nIR^2\,du}{2(R^2+u^2)^{3/2}}.$$$$\boxed{B(0)=\mu_0nI\left(1-\frac g{\sqrt{g^2+4R^2}}\right)}.$$

(b) Interpretation and further calculation

$$B(0)=\tfrac12\mu_0nI\Rightarrow\frac g{\sqrt{g^2+4R^2}}=\frac12\Rightarrow3g^2=4R^2.$$$$\boxed{g=2R/\sqrt3=1.15470R}.$$

Physical check

As g tends to zero the field approaches the uninterrupted solenoid value; for a very wide gap it tends to zero.