Power

Power: Definition and Mathematical Representation

Introduction

Power in physics is the rate at which work is done or energy is transferred. It quantifies how quickly energy is used, converted, or transmitted over time. Power is a scalar quantity and is crucial in both mechanical and electrical systems.

Definition

Mathematically, power is defined as the work done per unit time:

$P=\frac{W}{t}$

Where:

• $P$ is the power,
• $W$ is the work done,
• $t$ is the time interval.

In terms of energy transfer:

$P=\frac{E}{t}$

SI Unit

The SI unit of power is the watt (W), defined as:

$1\mathrm{W}=1\mathrm{J}/\mathrm{s}$

Other common units include:

• Kilowatt (kW): $1\mathrm{k}\mathrm{W}=1000\mathrm{W}$
• Horsepower (hp): $1\mathrm{h}\mathrm{p}\approx 746\mathrm{W}$

Instantaneous Power

When force and velocity are involved, instantaneous power is:

$P=\overrightarrow{F}\cdot \overrightarrow{v}$

Where:

• $\overrightarrow{F}$ is the force vector,
• $\overrightarrow{v}$ is the instantaneous velocity vector.

Average Power

The average power over a time interval is:

$P_{\text{avg}}=\frac{\mathrm{\Delta }E}{\mathrm{\Delta }t}$

This form is useful for systems with variable power consumption or delivery.

Power in Electrical Systems

In electrical circuits, power is defined as:

$P=IV$

Where:

• $I$ is the electric current,
• $V$ is the voltage.

Using Ohm's Law ($V=IR$), other forms include:

$P=I^{2}R=\frac{V^2}{R}$

Efficiency

Power efficiency is the ratio of useful output power to input power:

$\eta =\frac{P_{\text{out}}}{P_{\text{in}}}\times 100\mathrm{\%}$

Applications

• Power output of engines and motors
• Energy consumption in electronics
• Mechanical systems (e.g., lifting systems, turbines)
• Renewable energy (solar panel output, wind turbines)

See Also

• Energy
• Work (physics)
• Watt
• Electric Power
• Efficiency