Energy

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Energy is perhaps the single most useful concept in physics. Finding the energy of a system frequently yields a great deal of information about it; one may find its equilibrium position, its equations of motion, and calculate various relevant forces, all from basic considerations of the system's energy.

Most generally, energy is defined as the ability of a system to do "work". Work, in turn, is the product of a force and the distance over which it acts. For example, the work done by a force of 1 Newton over a distance of 1 meter (perhaps accelerating a mass) is 1 Joule. Indeed, this is one definition of a joule - the energy required to provide a force of 1 newton over a distance of 1 meter. There are many other definitions, corresponding to the various "forms" of energy - potential, kinetic, electrical, heat, chemical, etc. In a given system, energy may be transferred between any of these forms, but the sum of all these "types" of energy must always remain the same - this is the conservation of energy (see below).

Potential Energy

Potential energy is "stored" energy - it is energy that has been "put into" a system by doing work against some force acting on the system. For example, if a mass of 1kg on earth is lifted up off the ground by a distance of 1 meter, the mass has gained a potential energy of (1kg)(9.8 N)(1 meter) = 9.8 joules. This means that 9.8 joules of work have been done to lift the mass through a distance of 1 meter, against the force of gravity pulling the mass towards the earth.

Kinetic Energy

Kinetic energy is energy of motion - any mass that is moving at some velocity has a kinetic energy. Kinetic energy is given by $\begin{matrix} \frac{1}{2} \end{matrix} mv^2$ , where m is the mass of the object and v is its speed. For example, a 1kg mass moving at 2 m/s has 2 joules of kinetic energy.

Conservation of Energy

According to the first law of thermodynamics, energy is conserved within a closed system; i.e., it cannot be created or destroyed. If at some time a system has a certain total energy, then it must have that same total energy at all other times, provided that the system is isolated and cannot exchange energy with any other system.

A more general version of this law is the law of conservation of mass-energy, which says that the total amount of energy in a closed system, and its mass equivalent given by $E = m c 2$ is conserved. Mass can be turned into energy, and vice versa, but neither can be created from nothing or completely destroyed. This concept follows from the special theory of relativity - in classical cases (those in which things move much slower than the speed of light), it is not needed.

Illustration of Conservation of Energy

According to the law of conservation of energy, the total energy of a system must be constant. To illustrate this, consider a 1 kg mass that has been lifted up 1 meter above the surface of the earth. This gives it a potential energy of 2 joules. If the lifter now drops the mass, how fast will it be going when it hits the earth? By the conservation of energy, its total energy - in this case, the sum of its potential and kinetic energies - must remain constant. Since it had 2 joules of potential energy initially, and has none when it hits the earth (since it is 0 meters above the earth), the 2 joules of potential energy must all be transformed into kinetic energy. Solving the equation $2 J = \frac 1 2 \times 1 kg \times v^2$ , we find that v = 2 m/s. This same procedure may be followed to equate any of the forms of energy, so long as it is known into which forms the energy is being transferred, and by how much.

Relative Sizes of Energy Units

As discussed above, one joule is one Newton through one metre. One Newton is about the weight of an apple (100g ish). So, if you drop an apple on your foot from a height of one metre (so that the gravitational force acts on the apple through a distance of 1 meter) the energy dissipated when the apple hits the ground is 1 joule. (a typical site boot will protect you from a 300J impact).

One joule of electrical energy is defined to be one volt times one amp times one second. One AA battery might store 2.5 amp-hours at a little over one volt, which means the total energy capacity is 2.5 x 3600 x 1.1 = 10 kiloJoules.

Energy and Power

Energy is the total amount of work. Power is the rate at which the work is done with respect to time.

These scientifically correct terms can sometimes be confusing compared to the everyday uses of them. For instance, who has more energy; a world champion sprinter, or an 18 stone couch potato?

According to the scientific definition, the couch potato has more energy, because he has lots of chemical energy stored in the form of fat. If you burnt him, he would keep you warm for longer. What the sprinter has more of - what enables him to run 200 metres in less time than it takes to find the remote - is power.

Free Energy

If you are under 21, and living at home or at college, then chances are the power coming out of your wall socket is free, in that someone else is paying for it. That's the only kind of free energy there is, and it's more properly called "free to me" energy. Building a system to continuously output energy with no input is not possible. It doesn't matter whether it's called an over-unity machine, perpetual motion machine or what, it's simply not possible; it violates the law of conservation of energy, which is probably one of the most well-confirmed physical laws in existence. This is not scientific fascism, it's just the way it is. You don't have to believe in the conservation of energy, but if you act as if you do, then all your sums will come out right.

The first law of thermodynamics says that it is impossible to create or destroy energy. Any hypothetical machine that breaks this law is called a perpetual motion machine of the first kind - one that runs by creating energy.

The second law of thermodynamics says that disorder, entropy, always increases in a closed system, which means that it's impossible for a machine to run with 100% efficiency. A perpetual motion machine of the second kind would be one that runs forever by being 100% efficient and recycling all its energy perfectly.

Forms of Energy

Energy comes in many different forms.

Energy is often defined as the ability of a system to do work. Work is in turn defined most fundamentally as a force exerted over a distance. Anything that can be shown as equivalent to this is also work. The amount of available energy in a system is equal to the amount of work it can do.

How do you find the energy of a system? For every thing in that system that can store energy in some form, use the appropriate formula to compute how much there is of each form. Formulae for most forms are given below.

Energy stored (SI units throughout)
Type of energy	Formula	comments
kinetic energy	$\frac{1}{2}mV^2$	also $\frac{1}{2}$ momentum x velocity
gravitational potential	mgh	g = approximate potential
electrostatic energy	$\frac{1}{2}E^2\epsilon_0$	per m3
magnetic energy	$\frac{1}{2}B^2/\mu_0\ per\ m^3$	also $\frac{1}{2}BH$ also $\frac{1}{2}H^2\mu_0$

Work done (SI units throughout)
Type of energy	Formula	comments
linear mechanical	fd	force x distance
rotary mechanical	TW	torque x angle turned (radians)
gravitational potential	mgh	gh = approximate potential
electrical potential	QV	charge times voltage

Gravitational Potential

If a ball is held up and let go, it falls. Its falling does work on the ball - it accelerates - which makes kinetic energy out of what is called gravitational potential energy.

Electrical

Magnetic

Short cuts to calculations using energy concepts

What voltage does my Tesla coil generate

Regardless of the number of turns, and the coupling, and the resonant frequency, and the losses and the ..., remember that energy out = energy in.

Input energy - is stored on the tank cap. Perhaps 20nF charged to 15kV. Output energy. At some point, all the energy of the system, if we are lucky, will be stored as voltage on the output capacitance, which is maybe 20pF. There will be losses, but if we ignore them for the moment, we will get a figure that is an upper bound, an answer that will not be exceeded by the real system.

Ignoring losses, the output energy equals the input energy, $\frac{1}{2} CV^2$ is the same for each capacitor. As the output C is 1000 times less than the tank, the output voltage will be $\sqrt{1000}\approx 31$ times more, or 31x15kV = 465kV. In practice it will be less than this, likely much less as all the losses deduct their bit of input energy. But it is a useful approximation which cuts through all of the tedious maths.

How can I design the airgap of an inductor

What is the mechanical force on my capacitor plates

Energy

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