The Electric Field Due to a Charged Disk — Question 6

Question 6

An annulus has a maximum axial field away from its plane

An annulus with inner radius a and outer radius $R=2a$ has $E(z)=Cz[(z^2+a^2)^{-1/2}-(z^2+R^2)^{-1/2}]$, $C=2\pi k\sigma>0$.

1. Differentiate E and solve for the positive maximum height.

2. Express the result as $z/a$, and use the endpoint limits and derivative signs to classify the stationary point.

Question 6 - Solution

An annulus has a maximum axial field away from its plane

(a) Method and calculation

$$\frac1C\frac{dE}{dz}=\frac{a^2}{(z^2+a^2)^{3/2}}-\frac{R^2}{(z^2+R^2)^{3/2}}.$$$$\frac{z^2+R^2}{z^2+a^2}=(R/a)^{4/3}.$$

(b) Interpretation and further calculation

$$t=(R/a)^{2/3},\quad z_*^2=\frac{a^2t^2}{t+1}\Rightarrow z_*/a=\frac{2^{2/3}}{\sqrt{1+2^{2/3}}}\approx.986858.$$$$E(0)=0,\quad E(\infty)=0,\quad E(z)>0\ (z>0).$$The derivative starts positive and has only one positive zero before becoming negative, giving the global maximum.

Physical check

The vanishing center field does not imply the annulus produces a weak field at every height.