Energy

**Energy** is a physical quantity and property of objects which can be transferred to other objects or converted into different forms. There are two main types: kinetic and potential.

The dimensions of energy are [mass][length]2[time]-2.

## Types

Kinetic energy is the energy that an object has due to its motion, and potential energy is the energy that an object has due to its position in a force field. The total energy of a system is the sum of these two, and without external forces being applied is always conserved.

## Calculation

In classical mechanics, the kinetic energy is equal to

1
2

m

v
2

{\displaystyle \frac{1}{2} mv^2}

 and the potential energy is equal to

−

∫

r

0

r

F

⋅

d

r

{\displaystyle - \int\_{ \mathbf{ r\_{0} } }^\mathbf{r} \mathbf{F} \cdot\,d\mathbf{r}}

.

r
0

{\displaystyle \mathbf{r\_0}}

 is an arbitary reference point, and is typically at zero or infinity depending on the behaviour of the force (usually, if the force increases with distance, such as a spring constant, then the reference point is at zero, and if the force decreases with distance, such as a gravitational coupling, then the reference point is at infinity).

For an inverse square force field

k

r
2

r
^

{\displaystyle \frac{k}{r^2} \hat{r}}

, such as a gravitational field (

# k
−
G
M
m

{\displaystyle k = -GMm}

) or an electric field

# k

Q
q

4
π

ϵ
0

{\displaystyle k = \frac{Qq}{4 \pi \epsilon\_0}}

), the potential at a point is given by

k
r

{\displaystyle \frac{k}{r}}

.

### Electromagnetism

In electromagnetism, over all space the total energy stored in an electric field made by assembling static charges is

ε
0

2

∭
Ω

E
→

2

d

V

{\displaystyle \frac{\varepsilon\_0}{2} \iiint\_\Omega \vec{E}^2 \mathrm{d}V}

 and the total energy stored in a magnetic field made by assembling static currents is

1

2

μ
0

∭
Ω

B
→

2

d

V

{\displaystyle \frac{1}{2 \mu\_0} \iiint\_\Omega \vec{B}^2 \mathrm{d}V}

.

These energies stored in the electric and magnetic field are from the potential energy gained when work is done against the electric and magnetic fields to move the charges or currents into place.

### Thermodynamics

In thermodynamics, the internal energy is a function of the thermodynamic variables (temperature T, entropy S, pressure p, volume V, chemical potential μ, and particle number N), with a small change in energy dU given by

d
# U
T
d
S
−
p
d
V
μ
d
N

{\displaystyle dU = T dS - p dV + \mu dN}

. The overall energy of the system can be found by picking an arbitrary reference point at which energy is zero, and then integrating this differential between this point and the current state of the system.

## Kardashev Scale

The Kardashev scale deals with power (the time derivative of energy). This means that integrating the power output of a civilisation over its lifespan gives an energy figure for that civilisation. Typical maximum values of this are:

* Type 1.0 - 1033 joules
* Type 2.0 - 1045 joules
* Type 3.0 - 1055 joules
* Type 4.0 - 1065 joules
* Type 5.0 - 1075 joules

## Related Quantities

* Power - time derivative
* Specific energy - mass derivative
* Surcell energy density - volume derivative
* Force - length integral