A short cylinder, of radius a and length L, carries a “frozen-in” uniform polarization P, parallel to its axis. Find the bound charge, and sketch the electric field (i) for L ≫ a, (ii) for L ≪ a, and (iii) for L ≈ a. [This is known as a bar electret; it is the electrical analog to a bar magnet. In practice, only very special materials—barium titanate is the most “familiar” example—will hold a permanent electric polarization. That’s why you can’t buy electrets at the toy store.]
Question 4.11: A short cylinder, of radius a and length L, carries a “froze...
\rho_{b}=0 ; \sigma_{b}= P \cdot \hat{ n }=\pm P (plus sign at one end—the one P points toward; minus sign at the other—the one P points away from).
(i) L ≫ a. Then the ends look like point charges, and the whole thing is like a physical dipole, of length L and charge P \pi a^{2}. See Fig. (a).
(ii) L ≪ a. Then it’s like a circular parallel-plate capacitor. Field is nearly uniform inside; nonuniform “fringing field” at the edges. See Fig. (b).
(iii) L ≈ a. See Fig. (c).
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