Gauss' Law and Coulomb's Law — Question 7

Question 7

A dipole has zero closed flux while transferring flux between hemispheres

A compact ideal dipole has moment $p\hat{\mathbf z}$. On a sufficiently distant centered sphere of radius R its radial field is $E_r=2kp\cos\theta/R^3$. Treat this leading dipole field as the model. Here $k=1/(4\pi\epsilon_0)$.

1. Integrate its total outward flux and relate the result to the total dipole charge.

2. Find the northern hemisphere flux and its dependence on R. Explain why zero whole-sphere flux does not force each open-surface flux to vanish.

Question 7 - Solution

A dipole has zero closed flux while transferring flux between hemispheres

(a) Method and calculation

$$\Phi=\frac{2kp}{R}\int_0^{2\pi}\!d\phi\int_0^\pi\cos\theta\sin\theta\,d\theta=0.$$$$Q_{\mathrm{dipole}}=+q-q=0.$$

(b) Interpretation and further calculation

$$\Phi_N=\frac{4\pi kp}{R}\int_0^{\pi/2}\cos\theta\sin\theta\,d\theta=\frac{2\pi kp}{R}.$$$$\Phi_S=-\Phi_N,\quad\Phi_N(2R)=\tfrac12\Phi_N(R).$$The two hemispheres exchange equal signed contributions even though each generally has nonzero flux.

Physical check

The open-hemisphere flux falls as $1/R$ because dipole field falls as $1/R^3$ while area grows as $R^2$.