Thakshashila: Created page with "= 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,..."

2025-05-23T06:54:47Z

Created page with "= 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: <math> P = \frac{W}{t} </math> Where: * <math>P</math> is the power,..."

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= 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:

<math>
P = \frac{W}{t}
</math>

Where:
* <math>P</math> is the power,
* <math>W</math> is the work done,
* <math>t</math> is the time interval.

In terms of energy transfer:

<math>
P = \frac{E}{t}
</math>

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

<math>
1\, \mathrm{W} = 1\, \mathrm{J/s}
</math>

Other common units include:
* Kilowatt (kW): <math>1\, \mathrm{kW} = 1000\, \mathrm{W}</math>
* Horsepower (hp): <math>1\, \mathrm{hp} \approx 746\, \mathrm{W}</math>

== Instantaneous Power ==
When force and velocity are involved, instantaneous power is:

<math>
P = \vec{F} \cdot \vec{v}
</math>

Where:
* <math>\vec{F}</math> is the force vector,
* <math>\vec{v}</math> is the instantaneous velocity vector.

== Average Power ==
The average power over a time interval is:

<math>
P_{\text{avg}} = \frac{\Delta E}{\Delta t}
</math>

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

== Power in Electrical Systems ==
In electrical circuits, power is defined as:

<math>
P = IV
</math>

Where:
* <math>I</math> is the electric current,
* <math>V</math> is the voltage.

Using Ohm's Law (<math>V = IR</math>), other forms include:

<math>
P = I^2 R = \frac{V^2}{R}
</math>

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

<math>
\eta = \frac{P_{\text{out}}}{P_{\text{in}}} \times 100\%
</math>

== 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]]

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