Just like energy, we tend to hear the term power a lot. That is because, in our everyday life, power has a lot of meanings. However, when you isolate power into physics alone, you get one specific meaning, and that is: Power is the measurement of the rate at which we execute work (or, in a similar case, the rate at which energy is efficiently transferred from one location/place to another).
The ability to measure power accurately was one breakthrough that helped early engineers invent steam engines which brought about the industrial revolution. However, up until today, it remains essential to learn and understand the uses of power and how to utilize it in the most efficient way possible.
The Power Formula
Watts (W) is the standard unit used in the measurement of power. The unit got its name after a great Scottish industrialist and inventor known as James Watt; you can guess why the name watt. In the industrial era, we use the term “watts” in many objects, from machinery to home appliances and gadgets. You have most probably come across this term more than once, no doubt about it. For example, power outputs of appliances such as bulbs are generally advertised in watts hence chances of hearing this term goes further up.
- One watt = one joule of work done per second.
Hence, to measure power denoted as P in watts, we have to find energy change (ΔE) after execution of a certain job, then find the time taken to execute the job (Δt). With these two values in our possession, we can now calculate power using the following power formula:
P = ΔE/Δt
Have you ever heard of the term horsepower? Well, this term is also used as a unit for power, not as a substitute for watts, do not get mixed up on that. Horsepower is usually abbreviated as hp and gains its origin from the 17th century. In that time and age, horsepower referred to, you guessed it right, the power of an actual horse. Well, not typically the power of a horse as per their body, but the power of a horse when in use to turn a capstan. Since then, the metric horsepower (hp) has been in use as a measure for power.
Logic behind Horsepower
We can define horsepower as the amount of power required to lift seventy-five kilograms of mass across a distance of one meter in one second. People from the 17th century were able to measure the power of a horse efficiently using this method. But how many watts are in one horsepower?
Well, using our standard power formula, P = ΔE/Δt, we can easily get how much watts are in one horsepower as follows:
Since we know that when lifting against gravity, gravitational force affects the mass whereby it gains gravitational potential energy (Ep = m * g *h), hence, putting this into numbers, we get:
(75 kg * 9.807m/s2 * 1m)/ 1s = 735.5 watts
How can we Measure Power that is Varying?
In so many occasions whereby energy is being utilized, the rate of utilization varies over time. It might start at a high rate, go down, streamline for a while and then go up again. Take, for instance, the power consumption of a house (or home): During the day, most houses have minimal power consumption because all the lights are mostly off and grown-ups are usually off to work, and kids are in school, so no televisions are on too. However, at night every bulb is usually turned on, and the kids are busy watching tv while grown-ups are preparing meals. You can hence see clearly that at night, power consumption goes pivoting up, while during the day, there is minimal power consumption. That is what we basically call varying power consumption.
There are at least three separate ways in which we can express power in the situation named above:
- Instantaneous power (Pi)
- Average power (Pavg)
- Peak power (Ppk)
Electricity companies have to keep track of these three power expressions.
Refers to power measured at a specific given instant of time. Considering the power equation, P = ΔE/ Δt, instantaneous power is the measurement gotten when a change in time is pretty small, actually extremely small. If you plot a graph of power VS time (P vs. t), then the instantaneous power would be the value you read from the graph plot at any given time instance.
Refers to power measured over an extended period, i.e., when the change in time in the power equation is long. One method you can implement to calculate average time is finding the area under a power vs. time curve (which provides the total amount of work done) and dividing this value by the time taken. To get the best answer, you should solve this problem using calculus. However, you can still get an estimated answer using just geometry.
Refers to the most significant/maximum value that the instantaneous power can take in a specific system over an extended period. Stereo systems and car engines are great examples of systems that can deliver peak power that is way higher than their power average rating. However, to avoid damage, this peak power is only maintained for a short period. Nevertheless, in the systems named above, we consider high Ppk more important to the listening and driving experience more than a high Pavg.
Power Formula Physics extended
The power equation connects the work done with the time used to do the actual work. Since we already know that work is executed using force, and force can ultimately move objects, we can take the concept of power even further. Using the power equation, we learn a couple of things about the movement of a body over a specific amount of time. All we need to do is manipulate the equation, and the output will amaze you.
By substituting the amount of work executed by a specific force, say W = F*Δx cos x, into the power equation, we get:
P = W/ Δt being equal to P = (F* Δxcos θ)/ Δ
If, in this case, the force lies along the motion’s direction, then cos θ = 1, you can then rewrite the equation to:
P = F *v
Since we refer to the change in distance over a given period as velocity, we can write Pi = m * a * v.
Note that in the above-given equation, Pi = m * a * v, we have specified P as instantaneous; this is because the equation has both velocity and acceleration; therefore, velocity is changing over some time. All this can only make sense if we record and use velocity at a given instance. Otherwise, we shall need to implement average velocity as follows:
Pavg = m * a * ½(vfinal + Vinitial)
Using this equation, you can attain some pretty beneficial results. Take, for instance, a car with a mass of 1000 kilograms and as advertised, to the wheel, power output of up to 75 kilowatts (that is about a hundred horsepower). Furthermore, let’s say the advertiser claims that this car can gain constant acceleration over the range of 0 – 25 m/s.
By implementing the above information, we can easily calculate the time the car should use while in ideal condition to accelerate from 0 to 25 m/s.
Pavg = m * a * ½ v final
Since acceleration is Δv/ Δt:
Average power = m * (vfinal /t) * ½ v final
= (mv2final )/2t
You can then rearrange this to get:
t = (v2final * m)/2*Pavg
= ((25m/s)2 * 1000 kilograms)/ 2.7000 W
In real life, this acceleration is practically impossible because we have to consider the work occurring in the opposite direction (the negative work). Negative work occurs when the car gradually pushes air aside to move in the forward direction.
Dimensional Formula of Power Overview
Before we get into the dimensional formula of power, let us first go through the dimensional formula and its uses to understand the dimensional formula of power concept better.
The Dimension Formula
The dimensional formula of a physical quantity simply refers to that expression representing which base quantities have been included in the quantity and how.
We use the dimension formula to check the correctness of a formula dimensionally. It is also required in the conversion of systems from a particular unit to another. The dimensional formula is also used to explain how we express physical quantities in terms of fundamental units.
For this formula to work, there must be a multiplication-type dependency. Therefore, the derivation of formulas containing logarithmic, exponential, and trigonometric functions is impossible using this given method. Furthermore, formulas having more than a single term subtracted or added, for example (S = ut + ½ at 2), we cannot derive this by implementing the dimensional formula.
Dimensional Formula of Power
Up to this point, we based our discussion on the power equation (P = ΔE/ Δt). But now, let us switch our focus to another formula known as the dimensional formula of power.
The dimensional formula is as below:
M1 L2 T-3 whereby:
M represents mass
L represents length
T represents time
But how did we get to this absurd equation? Well, here is how:
Let’s say that Power (P) = Work x time – 1 = Joule x second -1 ……. (Step 1)
Since, Work = Force (M x a) x displacement = M1 L2 T-2L
Hence, the dimensional work formula = M1 L2 T-2…… (Step 2)
By substituting equation two into equation one, we get:
Power = Work x time – 1
Or P = [M1 L2 T-2] x [T-1] = M1 L2 T-3
We can hence conclude that the dimensional power formula is equal to:
M1 L2 T-3
Dimension simply refers to the depth, width, and height of objects or things. We refer to the amount of measurable space as dimensions.
The dimension formula simply reveals which and how many fundamental qualities are available in a physical quantity. So it is a pretty essential aspect of measurement and units.
While on the topic of power, what comes to mind first, well without even reading your mind, we can already tell that you are thinking about electricity. Well, that is because electricity is considerably connected to power; some people even refer to electricity as power. So, to ensure we do not leave any stones unturned, let us look at electricity and its formula.
*Electricity and its Formula
Electric energy stands as one of the most crucial scientific concepts. However, it is still pretty frequently misunderstood by many. So what is electric energy? And what formulas do we use in the calculation of electric energy?
To start us up on electricity, let us first look at electric energy before advancing to formulas used to solve for electric power.
Electric energy is energy we derive from the kinetic energy of charged particles or electric potential energy. Basically, we refer to this kind of energy as energy converted from electric potential energy or energy generated from the movement of electrons from one location/point to another.
Charged particles moving along or through a medium constitutes electricity or current.
Electrical Energy Formula
Every cell has two terminals, namely the negative and the positive terminal. The negative terminal contains electrons in excess, while the positive terminal is deficient in terms of electrons. Now, let us use A to represent the positive terminal; hence the electrical potential of this terminal becomes V (A). Similarly, let’s describe the negative terminal using B, whereby electrical potential becomes V (B). With this in mind, we can now define the flow of electric current as V (A) > V (B); this implies that current flows from A to B.
The potential difference between A and B is hence given by:
V = V (A) – V (B) > 0
The potential energy of charge denoted by Q at A is simply Q V (A), and at B, it’s Q V (B). Therefore, we calculate change in potential energy as:
∆Upot = Finalpot – Initialpot
= ∆Q[(V (B) – V (A)] = – ∆QV
= -I V ∆t (since I = ∆Q/ ∆t)
Taking into account the system’s kinetic energy, it could also change if the charges contained inside the conductor moved with no collision; this maintains the system’s total energy. When we convert the total energy, we get:
∆k = – Upot
Or ∆K = I V ∆ t > 0
Therefore, in the electricity field, whenever charges move freely across conductors, there is an increase in kinetic energy as they move.
When the moving charges collide, the energy gained by both of them is shared between their atoms. However, the collision also increases the atom’s vibration, which ultimately increases the amount of heat generated by the conductor. Therefore, some energy is usually dissipated in the form of heat in actual conductors.
Electric Energy Units
Electricity uses two basic measurement units that are:
- Watt – per second
When a single ampere of current passes/flows through a circuit for one second with the potential energy of one volt applied across it, this is what we term as one joule of energy.
Kilowatts per hour is the standard commercial unit for electrical energy, also termed the B.O.T. (Board of Trade unit).
1 kWh = 1000 x 60 x 60 watt-second
1 kWh = 36 x 105 joules or Ws
Here is something to note: when we talk about one unit, we are basically talking about one kilowatt per hour.
Examples of Electrical Energy
- A car battery – a chemical reaction occurs which leads to the formation of electrons that possess the energy to travel in an electric current. The moving charges generate electrical energy that the car’s circuit uses.
- Lighting – lighting is basically an electric discharge from the atmosphere.
Electric energy conversion to mechanical energy
Through different procedures and manipulations, we can convert electric energy into various forms of energy, for example:
- Heat energy
- Light energy
- Motion et cetera
Here are some examples of electric energy to mechanical energy conversions:
- Fans – fans contain motors that convert electric energy into mechanical energy
- Bulbs – bulbs convert electric energy into light energy
Now that we know all about electricity, how about we take things up a notch by connecting electricity to power.
Electric power refers to the rate at which energy in an electric circuit undergoes transformation or the rate at which execution of work occurs. Basically, electric power is a measure of the amount of energy utilized over a given period.
In physics, electric power refers to the transfer rate of electrical energy by a circuit per unit of time. In this case, energy can refer to:
- Potential energy
- Kinetic energy
In most cases, potential energy is usually considered (potential energy refers to the energy stored because of the electric field’s or charged particle’s relative position). We measure electric power in Watts, and is denoted as P.
|Vector or scalar||Scalar quantity|
|S.I. unit||Joules, watts per second|
|Power formula electricity-based||P = VIWhereby, I refers to the electric currentV refers to the potential difference (volts)|
Earlier on, we discussed the energy dissipated because of the heating up of conductors. In a time interval of ∆t, energy dissipated becomes:
∆W = I V ∆t
The power dissipated is actually energy dissipated per unit time, which is ultimately given by P = ∆W/ ∆t. however, we do know that the power equation is:
P = IV
With this in mind and the Ohms law (V = I.R.), we can make some manipulations to get:
P = I2R or P = V2/R
If you ever wondered why coils heat up, well, there is your answer. The power you gain via this formula is responsible for the production of heat and light.
Electricity Formulas extended
Well, electricity formulas go further and get more defined to bring to light more details on this incredible energy. Here are some of the most commonly used formulas for electricity.
Current refers to the flow of electric charge carriers, usually electron-deficient atoms or electrons.
We can calculate it using the following formula:
I = Q/t whereby,
Q = charge
t = time taken
We measure current in amperes (A).
Voltage refers to the pressure that comes from an electric circuit’s power source, which pushes charged electrons through a conducting loop. In doing so, it enables these charged electrons to do work such as producing heat and illuminating light.
In the early days, people referred to voltage as electromotive force or emf. That is why even in certain equations like the ohms law, we denote voltage with the symbol E.
To get the voltage of electric energy, you can employ the following formula:
V = W/Q
V = E/Q
W = work done
E = Energy
Units of measurement for voltage is volts (V)
Resistance refers to the measure of current flow opposition in a given electric circuit. All materials, at some point, resist the flow of a current. Hence, we separate materials into two broad categories according to their resistance, namely:
- Conductors – These types of materials offer minimal resistance. They hence transfer electric currents pretty well. Examples of conductors include aluminum, gold, copper, and solver.
- Insulators – these materials present pretty high resistance; they hence restrict current flow. Examples of insulators include plastic, wood, glass, paper, rubber.
To gain resistance of a certain object employ you can use the following formula:
R = ρI /A where,
ρ = resistivity
A = Area
I = length
R = V/I
The units of measurement for Resistance are in ohms (Ω).
Electric conductivity is the measure of how electric current navigates within a substance. Also, high conductivity within certain materials equals to high current density for a given applied Ppot (potential difference).
To get the conductivity of a material, you can employ the following formula:
Sigma = 1/ ρ
The unit of measure for conductivity is Siemens per meter or S/m.
We implement these electricity formulas to calculate unknown electrical parameters from a known electric circuit. Each formula is unique and provides essential information.
Power is a term that we basically hear almost every single day. If you work in an electrical or electronics-based company, then it is a term that you most probably use every single day. Over time, people tried to explore electricity, leaving a trail of formulas that helped us know more about power. We have discussed the main formulas for power plus other branches of the main formulas that come with it. Along with power, we have discussed the very crucial scientific concept known as electricity.
Based on modernization, electricity has become one of the most used resources in the world. Scientists and researchers have also dawned upon this concept whereby they have also left a trail of formulas, each unique but intertwined they produce great information about electricity. The best thing about power and electricity is that, they only get better, hence we hope that this article has given you a great boost in this incredible topics and left you anxious to learn more.