power
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What Is Power?
In the field of physics, power is the rate at which work on something is performed. This is also understood as the rate at which energy of one kind is converted to energy of another kind. Looking at this mathematically, power is taken to be:
P = W/t
where P is the power, W is the work, and t is interval of time in which the given amount of work is being done. The SI unit for power is the watt, or joule per second. As said above, power is also taken to be the rate at which energy is converted from one form to another. Mathematically, this appears as:
P = E/t
where E is the amount of energy converted. In mechanics, work is related to force in the following way:
W = F.Δd
Where F is the force on some object over a net displacement of Δd. This assumes that the motion given by Δd will always be along the same axis as the force acting on the object. If a force opposes the motion that gives a displacement Δd, that force will generate negative work, and if a force is perpendicular to the direction of motion, zero work will be done. Now we can take the same equation above and differentiate it with respect to time to get:
dW/dt = F.d(Δd)/dt = F.v = P
The time derivative of work is power, which is the same as a force multiplied by velocity (for some given instant, t). This is also known as the instantaneous power. If we have a given interval of time, a constant force, and a velocity, we can find the average power given by:
P_{average} = (1/Δt). ∫ F.v dt
where Δt is interval of time the force is applied to the object. This equation is useful when describing engines. The power an engine can produce is equal to the force it can exert on a vehicle multiplied by its velocity.
Electrical power is a function of current multiplied by velocity with respect to time so that:
P = I.V
This is another equation for instantaneous power. I(t) is a current at a given time and V(t) is the voltage for the same moment in time. If we are working with a resistor, we can find the power over that resistor using Ohm’s law:
P = I^{2}.R = V^{2} / R
For any given resistor, the power at a given time will be equal to the voltage squared at that time divided by the resistance. There are many more applications of electrical power, both in AC and DC circuits. For example, if one wanted to find the average power for a given AC circuit, we would use this equation:
P = (1/T).∫ (i.v) dt
where T is period of time in question (similar to Δt), i is the rms (root mean square) current, and v is the rms voltage. Because AC power is not constant, we use the rms current and voltage to get an average that is equivalent to
that found in a DC powered circuit.