Question

A particular piezoelectric device has a cross sectional area of \(10^{-5} \mathrm{~m}^{2}\) and an efficiency of \(5 \% .\) When a stress of \(1640 \frac{\mathrm{N}}{\mathrm{m}^{2}}\) is applied to the device, it oscillates with an average velocity of \(0.01 \frac{\mathrm{m}}{\mathrm{s}} .\) Calculate the power that can be generated from the device.

Short answer

The power generated is \(0.0000082\, \text{W}\).

Step-by-step solution

Calculate the Force

To begin, calculate the force applied to the device using the formula for stress: \(\text{Stress} = \frac{\text{Force}}{\text{Area}}\). Rearrange this formula to solve for force: \(\text{Force} = \text{Stress} \times \text{Area}\). Substitute the given values: \(\text{Force} = 1640 \frac{\text{N}}{\text{m}^2} \times 10^{-5} \text{ m}^2 = 0.0164 \text{ N}\).

Calculate the Power Input

Next, calculate the power input using the formula: \(\text{Power} = \text{Force} \times \text{Velocity}\). Substitute the known values for force and velocity: \(\text{Power} = 0.0164 \text{ N} \times 0.01 \frac{\text{m}}{\text{s}} = 0.000164 \text{ W}\).

Calculate the Power Output

Calculate the power output using the device's efficiency: \(\text{Power Output} = \text{Efficiency} \times \text{Power Input}\). The efficiency is given as \(5\%\) or \(0.05\). Therefore, \(\text{Power Output} = 0.05 \times 0.000164 \text{ W} = 0.0000082 \text{ W}\).

Key concepts

Efficiency Calculation

Efficiency is a crucial aspect in examining how well piezoelectric devices convert mechanical energy into electrical energy. It tells us what percentage of the input energy is effectively turned into useful output energy. In this scenario, the efficiency is given as 5%, which indicates that only a small portion of the input energy is used productively.

To calculate efficiency, you can use the formula:

• Efficiency (\(\eta\)) = \(\frac{\text{Power Output}}{\text{Power Input}} \times 100 \)

This formula helps to understand the energy loss during the conversion process.

For example, if a device produces a tiny power output of 0.0000082 W from a power input of 0.000164 W, its efficiency would indeed be 5%. This emphasizes the importance of selecting high-efficiency components in applications where energy conservation is critical.

Power Output Calculation

Calculating the power output of piezoelectric devices involves understanding the relationship between the device's efficiency and power input.

Given the device's efficiency of 5%, we need to first find out how much power (\(P_{in}\)) is initially provided to the system. This is often done in conjunction with finding the power input.

The formula used is:

• \(P_{out} = \eta \times P_{in}\)

With this formula, you plug in the values: \(P_{out} = 0.05 \times 0.000164 \, \text{W} = 0.0000082 \, \text{W}\).

This output power represents the small amount of energy converted into usable electrical energy by the piezoelectric device. Understanding this can be key to designing devices where even minimal amounts of power are significant.

Stress and Force Relationship

The relationship between stress and force is fundamental in the study of piezoelectric devices. Stress is defined as the force applied per unit area and is measured in newtons per square meter (N/m²).

This relationship is given by:

• \( \text{Stress} = \frac{\text{Force}}{\text{Area}} \)
• Rearranging to find force: \( \text{Force} = \text{Stress} \times \text{Area} \)

In our example, the stress applied is 1640 N/m², with a device area of \(10^{-5}\) m², leading to an applied force: \(0.0164 \, \text{N}\).

Understanding this relationship helps in determining how much mechanical force is applied to a system, which is subsequently converted into electrical energy by the piezoelectric device.

This concept underpins the operation of these devices, as the stress applied directly affects the force generated, and thereby the electrical output.