Question

A short cylinder, of radius a and length L, carries a "frozen-in" uniform polarization $\mathbf{P}$, parallel to its axis. Find the bound charge, and sketch the electric field (i) for $\mathbf{L}\mathbf{\gg }\mathbf{a}$, (ii) for $\mathbf{L}\mathbf{\ll }\mathbf{a}$, and (iii) for $\mathbf{L}\mathbf{\approx }\mathbf{a}$. [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.]

Short answer

(a) In the event that $\text{L}\gg \text{a}$, the field will roughly resemble that of a physical dipole, with "point charges" of magnitude $\pm \mathrm{PA}$spaced $L$apart.

(b) The field between the top and bottom of the cylinder in the case $\text{L}\ll \text{a}$will roughly resemble that of a parallel-plate capacitor with surface charge densities $\pm P$on the top and bottom.

(c) Simply take a look at the preceding two situations; they both have surface charge densities $\pm P$on the top/bottom. About as the photo depicts, the field looks.

Step-by-step solution

Write the given data from the question.

Consider a short cylinder, of radius $a$and length $L$, carries a "frozen-in" uniform polarization $P$, parallel to its axis .

(a) Determine for L≫a.

Draw the circuit diagram of electric field for$\text{L}\gg \text{a}$ .

Figure 1

For$\text{L}\gg \text{a}$ .

In the event that $\text{L}\gg \text{a}$, the field will roughly resemble that of a physical dipole, with "point charges" of magnitude$\pm \mathrm{PA}$ spaced apart.

(b) Determine for L≪a.

Draw the circuit diagram of electric field for $\text{L}\ll \text{a}$.

Figure 2

The field between the top and bottom of the cylinder in the case $\text{L}\ll \text{a}$will roughly resemble that of a parallel-plate capacitor with surface charge densities$\pm P$ on the top and bottom.

(b) Determine for L≈a.

Draw the circuit diagram of electric field for $\text{L}\approx \text{a}$

Figure 3

Simply take a look at the preceding two situations; they both have surface charge densities$\pm P$ on the top/bottom. About as the photo depicts, the field looks.

The surface bound charge is the only bound charge present in all three scenarios since the polarisation is constant, ${\rho }_b=0$.