Question

A particular piezoelectric device has a cross sectional area of \(10^{-5} \mathrm{~m}^{2}\). When a stress of \(800 \frac{\mathrm{N}}{\mathrm{m}^{2}}\) is applied, the device compresses by \(10 \mu \mathrm{m}\). Under these conditions, the device can generate \(2.4 \cdot 10^{-9} \mathrm{~J} .\) Calculate the efficiency of the device.

Short answer

The efficiency of the device is 3%.

Step-by-step solution

Calculate the Work Done on the Device by the Stress

Work done is determined by the formula \( W = F \cdot d \), where \( F \) is the force applied and \( d \) is the displacement. First, calculate the force \( F \) using the stress \( \sigma = \frac{F}{A} \). From here, \( F = \sigma \cdot A = 800 \frac{\mathrm{N}}{\mathrm{m}^2} \times 10^{-5} \mathrm{~m}^2 = 8 \times 10^{-3} \mathrm{~N} \). Then calculate \( W = F \cdot d = 8 \times 10^{-3} \mathrm{~N} \cdot 10 \times 10^{-6} \mathrm{~m} = 8 \times 10^{-8} \mathrm{~J} \).

Calculate the Efficiency of the Device

Efficiency is given by \( \eta = \frac{E_{out}}{E_{in}} \times 100\% \), where \( E_{out} \) is the energy generated and \( E_{in} \) is the work done. Substituting the known values \( E_{out} = 2.4 \times 10^{-9} \mathrm{~J} \) and \( E_{in} = 8 \times 10^{-8} \mathrm{~J} \), calculate \( \eta = \frac{2.4 \times 10^{-9} \mathrm{~J}}{8 \times 10^{-8} \mathrm{~J}} \times 100\% = 3\% \).

Key concepts

Efficiency Calculation

The efficiency of a piezoelectric device is a measure of how well it converts input energy into output energy. This metric is crucial for evaluating the performance of the device. To calculate efficiency, use the formula:

• \( \eta = \frac{E_{out}}{E_{in}} \times 100\% \)

Here:

• \( E_{out} \) is the energy generated by the device, which is the useful output.
• \( E_{in} \) is the energy input into the system, calculated as the work done on the device during operation.

In our case:

• \( E_{out} = 2.4 \times 10^{-9} \mathrm{~J} \)
• \( E_{in} = 8 \times 10^{-8} \mathrm{~J} \)

Substitute these values into the efficiency formula to find:

• \( \eta = \frac{2.4 \times 10^{-9}}{8 \times 10^{-8}} \times 100\% = 3\% \)

This means the device is converting 3% of the input energy into useful output energy, which is typical for piezoelectric devices. Despite the low efficiency, they are preferred for their reliability and their ability to convert mechanical stress directly into electrical energy.

Work and Energy

Understanding work and energy is fundamental in assessing the performance of piezoelectric devices. Work is done when a force causes an object to move in the direction of the force. In the context of piezoelectric devices, work performed on the device involves mechanical stress.The work done on the device is calculated using:

• \( W = F \cdot d \)

where:

• \( F \) is the force applied, which can be derived from stress and the device's cross-sectional area.
• \( d \) is the displacement the device undergoes.

To find work, initially calculate the force using the formula:

• \( F = \sigma \cdot A \)

Given:

• \( \sigma = 800 \frac{\mathrm{N}}{\mathrm{m}^{2}} \)
• \( A = 10^{-5} \mathrm{~m}^{2} \)

The force becomes:

• \( F = 800 \times 10^{-5} = 8 \times 10^{-3} \mathrm{~N} \)

With a displacement of \( 10 \mu \mathrm{m} \) or \( 10 \times 10^{-6} \mathrm{~m} \), the work done is:

• \( W = 8 \times 10^{-3} \mathrm{~N} \cdot 10 \times 10^{-6} \mathrm{~m} = 8 \times 10^{-8} \mathrm{~J} \)

This work is the total energy input into the device, an essential component for calculating efficiency.

Stress and Force Calculation

Piezoelectric devices rely on their ability to convert stress into electric energy. Calculating stress and subsequently force is crucial to understanding the energy conversion process.Stress, \( \sigma \), is defined as the force per unit area, and is calculated using:

• \( \sigma = \frac{F}{A} \)

In this exercise:

• Stress \( \sigma = 800 \frac{\mathrm{N}}{\mathrm{m}^{2}} \), which serves as the given input condition.
• The cross-sectional area of the device is \( A = 10^{-5} \mathrm{~m}^{2} \).

Given these, the force \( F \) can be calculated by rearranging the stress formula:

• \( F = \sigma \times A \)
• This results in \( F = 800 \times 10^{-5} = 8 \times 10^{-3} \mathrm{~N} \).

This force is the mechanical input acting over the device's area, which when combined with the displacement, allows for the calculation of the work done on the device. Understanding the relationship between stress, force, and area is foundational for mastering piezoelectric device operations.