Question

A piezoelectric material has permittivity \(\epsilon_{r}=2.5 .\) If the material is placed in an electric field of strength \(|\vec{E}|=2 \cdot 10^{3} \frac{\mathrm{V}}{\mathrm{m}}\) and is subjected to a stress of \(|\vec{\zeta}|=200 \frac{\mathrm{N}}{\mathrm{m}^{2}},\) the material polarization of the material is \(3.2 \cdot 10^{-8} \frac{\mathrm{C}}{\mathrm{m}^{2}} .\) Calculate \(d,\) the piezoelectric strain constant.

Short answer

The piezoelectric strain constant \( d = -6.125 \times 10^{-11} \frac{\mathrm{C}}{\mathrm{N}} \).

Step-by-step solution

Understand the given formula for polarization

The total polarization \( P_t \) in a piezoelectric material subjected to an electric field \( E \) and mechanical stress \( \zeta \) is given by \( P_t = \epsilon_0 \epsilon_r E + d \zeta \), where \( \epsilon_0 \) is the permittivity of free space, \( \epsilon_r \) is the relative permittivity, \( d \) is the piezoelectric strain constant, \( E \) is the electric field, \( \zeta \) is the stress, and \( P_t \) is the total polarization.

Substitute known values into the formula

We know that \( P_t = 3.2 \cdot 10^{-8} \frac{\mathrm{C}}{\mathrm{m}^{2}} \), \( E = 2 \cdot 10^3 \frac{\mathrm{V}}{\mathrm{m}} \), \( \epsilon_r = 2.5 \), \( \epsilon_0 = 8.85 \times 10^{-12} \frac{\mathrm{C}^2}{\mathrm{N} \cdot \mathrm{m}^2} \), and \( \zeta = 200 \frac{\mathrm{N}}{\mathrm{m}^2} \). Substitute these into the equation: \( 3.2 \times 10^{-8} = (8.85 \times 10^{-12} \times 2.5 \times 2000) + d \times 200 \).

Simplify and solve for \(d\)

Calculate \( \epsilon_0 \epsilon_r E \) first: \( 8.85 \times 10^{-12} \times 2.5 \times 2000 = 4.425 \times 10^{-8} \). Substituting back, we have \( 3.2 \times 10^{-8} = 4.425 \times 10^{-8} + d \times 200 \). Rearrange to solve for \( d \): \( d \times 200 = 3.2 \times 10^{-8} - 4.425 \times 10^{-8} = -1.225 \times 10^{-8} \). Thus, \( d = \frac{-1.225 \times 10^{-8}}{200} = -6.125 \times 10^{-11} \frac{\mathrm{C}}{\mathrm{N}} \).

Expressing the final answer

The piezoelectric strain constant \( d \) is calculated to be \(-6.125 \times 10^{-11} \frac{\mathrm{C}}{\mathrm{N}}\). This negative value indicates that the polarization opposes the stress-induced polarization within the material.

Key concepts

Polarization

In the context of piezoelectric materials, polarization refers to the creation of an electric dipole moment in response to an external electric field or mechanical stress. It is a crucial property that describes how much charge separation occurs within the material. When an electric field is applied to a piezoelectric material, the internal structure of the material rearranges, leading to polarization. This rearrangement causes a voltage across the material, making piezoelectric materials unique in their electro-mechanical interaction.

Polarization in piezoelectric materials is mathematically expressed as the total polarization, typically denoted as \(P_t\).

• It incorporates both the electric field-induced polarization and the stress-induced polarization.
• This can be summarized by the equation: \(P_t = \epsilon_0 \epsilon_r E + d \zeta\).

Understanding polarization helps in designing and utilizing piezoelectric materials effectively in applications such as sensors and actuators.

Relative Permittivity

Relative permittivity, also known as the dielectric constant \(\epsilon_r\), represents a material's ability to permit electric field lines to pass through it. It is the ratio of the permittivity of a material to the permittivity of free space, \(\epsilon_0\).

• A higher relative permittivity indicates that the material can better store charge when an electric field is present.
• In the case of our piezoelectric material, we have \(\epsilon_r = 2.5\).

Relative permittivity is key in determining how the material will interact with electric fields. This property is crucial for calculating the polarization due to the electric field within the material using the formula \(P_t = \epsilon_0 \epsilon_r E + d \zeta\).

In essence, the relative permittivity helps in predicting the effectiveness of a material in applications where it must respond to electric fields.

Electric Field

An electric field \(E\) influences charged particles, leading to effects such as polarization in piezoelectric materials. It measures the force per unit charge provided by an electric source.

• In our example, the electric field across the material is \(2 \times 10^{3} \frac{\text{V}}{\text{m}}\).
• This field interacts with the charges in the material, realigning them and creating an internal polarization.

The role of the electric field in piezoelectric materials cannot be overstated, as it influences the magnitude of the polarization. The efficiency and sensitivity of piezoelectric devices, such as ultrasound transducers, heavily depend on the ability of the material to react to the input electric field.

Mechanical Stress

Mechanical stress \(\zeta\) is the force applied to a material divided by its area. In piezoelectric materials, mechanical stress can lead to changes in electrical polarization.

When stress is applied, it can induce dipoles to form, adding to or opposing the electric field's influence on polarization.

• In our scenario, \(\zeta\) is defined as \(200 \frac{\text{N}}{\text{m}^2}\).
• Stress-induced polarization is comprehended through the piezoelectric strain constant \(d\), which indicates how much the polarization changes due to mechanical stress.

The consideration of mechanical stress is pivotal when designing piezoelectric sensors or actuators, as it impacts their durability and efficiency. By understanding how mechanical load changes the internal structure and charge distribution, engineers can optimize the materials for desired applications.