Applying Gauss' Law: Cylindrical Symmetry — Question 9

Question 9

An eccentric cylindrical cavity has a uniform field by superposition

An infinite insulating cylinder of uniform density $\rho$ contains an empty cylindrical cavity parallel to its axis. The cavity axis is displaced from the main axis by transverse vector $\mathbf d$, and the cavity lies entirely inside the original cylinder. Define $\mathbf r$ and $\mathbf r\prime=\mathbf r-\mathbf d$ from the two axes.

1. Model the cavity by superposing a cylinder of density $-\rho$ and derive the field at any point inside it.

2. Determine how the result changes when the cavity radius changes with $\mathbf d$ fixed. Explain why a Gaussian cylinder centered on the cavity alone is not enough to infer the field directly.

Question 9 - Solution

An eccentric cylindrical cavity has a uniform field by superposition

(a) Method and calculation

$$\mathbf E_{\mathrm{full}}=\frac\rho{2\epsilon_0}\mathbf r,\quad\mathbf E_{\mathrm{removed}}= -\frac\rho{2\epsilon_0}\mathbf r\prime.$$$$\mathbf E_{\mathrm{cavity}}=\frac\rho{2\epsilon_0}(\mathbf r-\mathbf r\prime)=\frac\rho{2\epsilon_0}\mathbf d.$$

(b) Interpretation and further calculation

$$\boldsymbol\nabla\mathbf E_{\mathrm{cavity}}=0,\quad |\mathbf E_{\mathrm{cavity}}|=\frac{|\rho|d}{2\epsilon_0}.$$$$\text{Changing cavity radius at fixed }\mathbf d\text{ leaves this field unchanged}.$$The full distribution is not cylindrically symmetric about the cavity axis, but each superposed complete cylinder has the needed symmetry.

Physical check

A concentric cavity has $\mathbf d=0$ and therefore zero interior field, as expected from cylindrical symmetry.