Question

A crystalline material is both piezoelectric and pyroelectric. When an external electric field of $|\overrightarrow{E}|=100\frac{\mathrm{V}}{\mathrm{m}}$ is applied, the material polarization is determined to be $|\overrightarrow{P}|=1500{ϵ}_0\frac{\mathrm{C}}{{\mathrm{m}}^2}$. When both a stress of $|\overrightarrow{\varsigma }|=30\frac{\mathrm{N}}{{\mathrm{m}}^2}$ and an external electric field of $|\overrightarrow{E}|=100\frac{\mathrm{V}}{\mathrm{m}}$ are applied, the material polarization is determined to be $|\overrightarrow{P}|=$ $6.0123\cdot {10}^{-6}\frac{\mathrm{C}}{{\mathrm{m}}^2}$. When a temperature gradient of $\mathrm{\Delta }T=50{}^0\mathrm{C},\mathrm{a}$ stress of $|\overrightarrow{\varsigma }|=30\frac{\mathrm{N}}{{\mathrm{m}}^2},$ and an external electric field of $|\overrightarrow{E}|=100\frac{\mathrm{V}}{\mathrm{m}}$ are applied, the material polarization is determined to be $|\overrightarrow{P}|=$ $6.3\cdot {10}^{-6}\frac{\mathrm{C}}{{\mathrm{m}}^2}.$ Find: \- The relative permittivity of the material \- The piezoelectric strain constant \- The magnitude of the pyroelectric coefficient

Short answer

The relative permittivity is approximately 1500, the piezoelectric strain constant is approximately $2.004\times {10}^{-7}\mathrm{C}/\mathrm{N}$, and the pyroelectric coefficient is approximately $5.754\times {10}^{-9}\mathrm{C}/\left({\mathrm{m}}^2{\cdot }^{\circ }\mathrm{C}\right)$.

Step-by-step solution

Determine Electric Displacement

The electric displacement $\overrightarrow{D}$ is related to polarization $\overrightarrow{P}$ and electric field $\overrightarrow{E}$ as $\overrightarrow{D}={ϵ}_0\overrightarrow{E}+\overrightarrow{P}$. Use ${ϵ}_0=8.854\times {10}^{-12}\mathrm{C}/\mathrm{V}\cdot \mathrm{m}$. For $|\overrightarrow{E}|=100\mathrm{V}/\mathrm{m}$ and $|\overrightarrow{P}|=1500{ϵ}_0{\mathrm{C}/\mathrm{m}}^2$, substitute into the formula:$$|\overrightarrow{D}|={ϵ}_0\times 100+1500{ϵ}_0$$$$|\overrightarrow{D}|=1500{ϵ}_0+8.854\times {10}^{-10}{\mathrm{C}/\mathrm{m}}^2$$

Calculate Relative Permittivity

The relative permittivity ${ϵ}_r$ is given by the formula $D={ϵ}_0{ϵ}_{r}E$. Rearrange to solve for ${ϵ}_r$:$${ϵ}_r=\frac{|D|}{{ϵ}_0|E|}$$Using $|D|\approx 1500{ϵ}_0$ and $|E|=100$,$${ϵ}_r\approx 1500$$

Calculate Piezoelectric Strain Constant

When the stress $|\overrightarrow{\varsigma }|=30\mathrm{N}/{\mathrm{m}}^2$ is applied, the difference in polarization must be attributed to the piezoelectric effect:$$\mathrm{\Delta }{P}_{\text{piezo}}=(6.0123\times {10}^{-6}-1500{ϵ}_0)\mathrm{C}/{\mathrm{m}}^2$$$$\mathrm{\Delta }{P}_{\text{piezo}}\approx 6.0123\times {10}^{-6}-8.854\times {10}^{-10}\approx 6.0114\times {10}^{-6}\mathrm{C}/{\mathrm{m}}^2$$The piezoelectric strain constant $d$ is given by $\mathrm{\Delta }{P}_{\text{piezo}}=d\times |\overrightarrow{\varsigma }|$:$$d=\frac{\mathrm{\Delta }{P}_{\text{piezo}}}{|\overrightarrow{\varsigma }|}\approx \frac{6.0114\times {10}^{-6}}{30}\approx 2.004\times {10}^{-7}\mathrm{C}/\mathrm{N}$$

Calculate Pyroelectric Coefficient

Considering a temperature gradient $\mathrm{\Delta }T={50}^{\circ }C$, the change in polarization due to the pyroelectric effect is given by:$$\mathrm{\Delta }{P}_{\text{pyro}}=(6.3\times {10}^{-6}-6.0123\times {10}^{-6})\mathrm{C}/{\mathrm{m}}^2$$$$\mathrm{\Delta }{P}_{\text{pyro}}=0.2877\times {10}^{-6}\mathrm{C}/{\mathrm{m}}^2$$The pyroelectric coefficient $p$ is given by $\mathrm{\Delta }{P}_{\text{pyro}}=p\times \mathrm{\Delta }T$:$$p=\frac{0.2877\times {10}^{-6}}{50}\approx 5.754\times {10}^{-9}\mathrm{C}/\left({\mathrm{m}}^2{\cdot }^{\circ }\mathrm{C}\right)$$

Key concepts

Pyroelectric Effect

The pyroelectric effect refers to the phenomenon where certain crystals produce an electric charge in response to changes in temperature. When the temperature of a pyroelectric material is altered, it results in a change in the polarization of the material.
This change in polarization creates a temporary voltage across the material, forming the basis of the pyroelectric effect. The pyroelectric coefficient, often represented as $p$, quantifies this effect and is defined by the change in polarization per unit change in temperature. In our example, the pyroelectric coefficient was calculated to be approximately $5.754\times {10}^{-9}\mathrm{C}/({\mathrm{m}}^2{\cdot }^{\circ }\mathrm{C})$.
Some applications of the pyroelectric effect include infrared sensors and fire detection systems, where slight temperature changes induce measurable charges within the crystal. This property makes pyroelectric materials highly valued in various sensing technologies.

Relative Permittivity

Relative permittivity, also known as the dielectric constant, is an essential material property in physics and materials science. It measures the ability of a material to store electrical energy in an electric field relative to the vacuum. This value is crucial in determining how an electric field affects and interacts with a dielectric material.
The equation relating electric displacement $\overrightarrow{D}$, electric field $\overrightarrow{E}$, and permittivity $ϵ$ is $D={ϵ}_0{ϵ}_{r}E$, where ${ϵ}_0$ is the permittivity of free space and ${ϵ}_r$ is the relative permittivity.
For the given exercise, the relative permittivity was found to be approximately $1500$. This indicates that when compared to a vacuum, the material can store significantly more electrical energy. This property is vital for designing capacitors and understanding how materials affect capacitance.

Electric Displacement

Electric displacement is a vector field $\overrightarrow{D}$ that appears in Maxwell's equations relating to the electric field $\overrightarrow{E}$ and polarization $\overrightarrow{P}$ of a dielectric material. It represents how an electric field interacts with the charge distribution within a medium. The formula to calculate electric displacement is $\overrightarrow{D}={ϵ}_0\overrightarrow{E}+\overrightarrow{P}$.
Understanding electric displacement is crucial for analyzing how materials with different permittivities affect the electric field. In our example, after substituting given values, the displacement was calculated which helped in determining the relative permittivity. This knowledge can be applied to areas such as material science for designing better insulators and conductors. It provides insight into how different materials respond to electric fields, which is integral in electronics and telecommunications.

Piezoelectric Strain Constant

The piezoelectric strain constant, often denoted as $d$, characterizes the piezoelectric effect. It describes how much electric charge is produced under mechanical stress or how much mechanical strain occurs under an electric field.
When mechanical stress is applied to a piezoelectric material, it results in polarization change. This change in polarization, compared to when only an electric field is applied, can be used to calculate the piezoelectric strain constant using the relation $\mathrm{\Delta }{P}_{\text{piezo}}=d\times |\overrightarrow{\varsigma }|$, where $\mathrm{\Delta }{P}_{\text{piezo}}$ is the change in polarization due to mechanical stress and $|\overrightarrow{\varsigma }|$ is the applied stress.
In our exercise, $d$ was found to be approximately $2.004\times {10}^{-7}\mathrm{C}/\mathrm{N}$. This constant is a fundamental property for materials used in sensors, actuators, and motors due to its role in translating mechanical and electrical energy.