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3.4: Problems

Last updated

May 22, 2022
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- 3.3: Notation Quagmire
- 4: Antennas

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3.1. For each of the three crystalline materials below

Find the crystal point group to which it belongs. (Hint: use http://www.mindat.org )
Using Table 2.3.1, determine whether or not the material is piezoelectric.
Using Table 2.3.1, determine whether or not the material is pyroelectric.
Using Table 2.3.1, determine whether or not the material is Pockels electro-optic.

(a) ZnS (sphalerite)

(b) HgS (cinnabar)

3.2. Cane sugar, also called saccharose, has chemical composition C $_12$ H $_22$ O $_11$ and belongs to the crystal point group given by 2 in Hermann-Maguin notation [38]. Reference [38] lists values specified in cgse units for its piezoelectric constant as $10.2 \cdot 10^{-8} \frac{\text{esu}}{\text{dyne}}$ and its pyroelectric coefficient as $0.53 \frac{\text{esu}}{cm^2 \cdot ^{\circ}\text{C}}$ . Convert these values to the SI units of $\frac{m}{V}$ and $\frac{C}{m^2\cdot K}$ respectively.

Hint: The electrostatic unit or statcoulomb is a measure of charge [7] where $1 \text{ esu} = 1 \text{ statC} = 3.335641 \cdot 10^{-10} C \nonumber$ and the dyne is a measure of force where $1 \text{ dyne} = 10^{-5} N$ .

3.3. A material has relative permittivity $\epsilon_{r\;x}$ when no external electric field is applied. The coefficient $\chi^{(2)}$ is measured in the presence of an external electric field of strength $|\overrightarrow{E}|$ . Assume that $\chi^{(3)}$ and all higher order coefficients are zero. Find the Pockels coefficient $\gamma$ as a function of the known quantities $\epsilon_{r\;x}$ , $\chi^{(2)}$ , and $|\overrightarrow{E}|$ .

3.4. The first figure below shows the displacement flux density $|\overrightarrow{D}|$ as a function of the strength of an applied electric field intensity $|\overrightarrow{E}|$ in a non-electro-optic material. The second figure below shows the displacement flux density $|\overrightarrow{D}|$ as a function of the strength of an applied electric field intensity $|\overrightarrow{E}|$ in a ferroelectric electro-optic material. The solid line corresponds to an unpoled material. The dotted line corresponds to the material after it has been poled in the $\hat{a}_z$ direction, and the dashed line corresponds to the material after it has been poled in the $-\hat{a}_z$ direction.

(a) For the non-electro-optic material, find the relative permittivity, $\epsilon_r$ . Also find the magnitude of the material polarization, $\overrightarrow{P}$ .

(b) Assume the ferroelectric electro-optic material is poled by a strong external electric field, and then the field is removed. Find the magnitude of the material polarization $|\overrightarrow{P}|$ after the external field is removed.

(c) Assume the ferroelectric material is poled in the $-\hat{a}_z$ direction by a strong external field, and then the field is removed. A different external electric field given by $\overrightarrow{E} = 100\hat{a}_z \frac{V}{m}$ is applied. Find the approximate relative permittivity of the material.

3.4.1.png

3.4.2.png

3.5. A crystalline material is both piezoelectric and pyroelectric. When an external electric field of $|\overrightarrow{E}| = 100 \frac{V}{m}$ is applied, the material polarization is determined to be $|\overrightarrow{P}| = 1500\epsilon_0 \frac{C}{m^2}$ . When both a stress of $|\overrightarrow{\varsigma}| = 30 \frac{N}{m^2}$ and an external electric field of $|\overrightarrow{E}| = 100 \frac{V}{m}$ are applied, the material polarization is determined to be $|\overrightarrow{P}| = 6.0123 \cdot 10^{-6} \frac{C}{m^2}$ . When a temperature gradient of $\Delta T = 50\; ^\circ C$ , a stress of $|\overrightarrow{\varsigma}| = 30 \frac{N}{m^2}$ , and an external electric field of $|\overrightarrow{E}| = 100 \frac{V}{m}$ are applied, the material polarization is determined to be $|\overrightarrow{P}| = 6.3 \cdot 10^{-6} \frac{C}{m^2}$ . Find:

The relative permittivity of the material
The piezoelectric strain constant
The magnitude of the pyroelectric coefficient

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