# Electrical power

As electrical energy ( symbols ) refers to energy , by means of electricity transferred or in electric fields stored is. Energy that is converted between electrical energy and other forms of energy is called electrical work (symbol ). Before 1970, the term electricity work was also in use. In the energy industry , the electrical energy transmitted is also called the amount of electricity or (less often) the amount of electricity . ${\ displaystyle E}$ ${\ displaystyle W}$

The watt-second ( symbol Ws) or the equivalent joule (J) is used as the unit of measurement for electrical energy and work . In the case of quantitative information on energy consumption in the field of electrical energy technology , the larger unit of measurement is the kilowatt hour (kWh).

1 kWh = 3,600,000 J; 1 J ≈ 2.778 · 10 −7 kWh.

Electrical energy can be used in many ways, as it can be converted into other forms of energy with low losses and can be easily transported. Their generation and supply to the economy and consumers is of great importance in modern societies.

## Manifestations

In power plants , batteries and accumulators , electrical energy is generated by converting other forms of energy, e.g. B. from thermal energy or chemical energy . This is transported to the consumers via power lines, where it is converted back into other types of energy (kinetic, potential, light or heat energy).

The electrical energy is localized in the electromagnetic field , which manifests itself macroscopically in current and voltage (see below ).

### Energy of a battery

A battery maintains the basis of their chemical energy content at sufficiently low current between its poles, a constant voltage upright (the voltage may decrease as the current strength increases). This happens until a certain charge has flowed through the circuit. How much charge can flow can be determined using the nominal capacity (common unit: ampere-hour , 1 Ah = 3600  C ). Then the voltage drops below its nominal value. According to the definition of electrical voltage , the work is done (see below), so that, for example, a mignon cell with 1.5 V nominal voltage and 2.3 Ah nominal capacity can provide at least 3.45 Wh ≈ 12 kJ of electrical energy. ${\ displaystyle Q}$${\ displaystyle U}$${\ displaystyle Q \ cdot U}$

### Field energy

Electrical energy can be stored both in the electrical field and in the magnetic field. This includes storing energy in a capacitor (electrostatic field) or in a coil (magnetic field).

Magnetic energy is expressed in a magnetic field and exerts a force on moving charges, the so-called Lorentz force . One also speaks of electromagnetism . Electromagnetic forces can be very strong; they are used in electric motors and generators . In practice, magnetic energy can be stored briefly in a coil; With superconducting magnetic energy stores , longer storage times with high energy are possible.

In an electrical oscillating circuit , electrical and magnetic energy are periodically converted into one another.

Due to the mathematical equality of energy and work , the formula symbols are used depending on their usefulness. In this section , although the equations about the field energy are also often noted in the literature , as it is used in the following section to avoid confusion with the electric field. ${\ displaystyle E}$${\ displaystyle W}$

Energy of a capacitor

The energy that is stored in the electric field of a capacitor is

${\ displaystyle E = {\ frac {1} {2}} \ cdot C \ cdot U ^ {2}}$,

where is the capacitance of the capacitor and the applied electrical voltage. ${\ displaystyle C}$${\ displaystyle U}$

Capacitors store significantly smaller amounts of energy than batteries. Double-layer capacitors are used for larger amounts of energy to be stored for which the use of a battery or accumulator is not an option .

Energy of a coil

The energy that is stored in the magnetic field of a coil is

${\ displaystyle E = {\ frac {1} {2}} \ cdot L \ cdot I ^ {2}}$,

where is the inductance of the coil and the strength of the current flowing through it. ${\ displaystyle L}$${\ displaystyle I}$

## Electrical work

The electrical work involved in moving a charge between two points between which the voltage exists is according to the definition of electrical voltage ${\ displaystyle Q}$${\ displaystyle U}$

${\ displaystyle W = U \ cdot Q}$.

When the charge moves against the electrical field forces, the electrical energy increases at the expense of other forms of energy (positive electrical work), while when the charge moves in the direction of the electrical field forces, the electrical energy decreases in favor of other forms of energy (negative electrical work). In calculations, these signs are only obtained if the physical sign conventions are observed; electrical voltages must be assessed as positive if the electrical potential increases in the direction under consideration .

### Work in the circuit

If the voltage and current strength are constant over a period of time ( i.e. equal quantities ), the charge can be replaced by the product of current strength and time period. The work in this period is: ${\ displaystyle \ Delta t}$${\ displaystyle \ Delta W}$

${\ displaystyle \ Delta W = U \ cdot I \ cdot \ Delta t}$.

The product of voltage and current strength is the electrical power , this indicates the work per period and is also constant under the specified conditions: ${\ displaystyle P}$

${\ displaystyle \ Delta W = P \ cdot \ Delta t}$.

For the electrical energy requirement of an electrical consumer that is operated with mains voltage , this is usually marked with its nominal power , often on a nameplate. The user defines the period of time by how long the consumer is switched on. (For devices with standby mode , in which only parts can be switched off and other parts run all day, the standby power is rather concealed.) If a consumer is operated differently with alternating quantities, its voltage drop and active current consumption must be known.

In the more general case of variable voltage and current , the instantaneous value of the power (because of and ) ${\ displaystyle u}$${\ displaystyle i}$${\ displaystyle p}$${\ displaystyle p = \ mathrm {d} W / \ mathrm {d} t}$${\ displaystyle i = \ mathrm {d} Q / \ mathrm {d} t}$

${\ displaystyle p = u \ cdot i}$,

the electrical work results from this through integration with respect to time:

${\ displaystyle W_ {12} = \ int _ {t_ {1}} ^ {t_ {2}} u \ cdot i \ cdot \ mathrm {d} t}$.

### Work in the electric field

The work involved in shifting a charge in an electric field from point A to point B is calculated as the scalar product of force and path , as in mechanics , in the more general case of non-constant force as the integration of the force according to the path: ${\ displaystyle {\ vec {F}}}$${\ displaystyle {\ vec {s}}}$

${\ displaystyle W_ {ab} = \ int _ {a} ^ {b} {\ vec {F}} \ cdot \ mathrm {d} {\ vec {s}}}$.

The force results as the counterforce to the electric field force on the charge, which is calculated as the product of electric field strength and charge: ${\ displaystyle {\ vec {E}}}$

${\ displaystyle {\ vec {F}} = - ({\ vec {E}} \ cdot Q)}$.

The electrical work can thus be expressed in general as:

${\ displaystyle W_ {ab} = - Q \ int _ {a} ^ {b} {\ vec {E}} \ cdot \ mathrm {d} {\ vec {s}}}$.

### Work when changing the distance between two charges

The force on a charge which is at a distance from a charge is according to Coulomb's law${\ displaystyle q}$${\ displaystyle r}$${\ displaystyle Q}$

${\ displaystyle F _ {\ text {C}} = {\ frac {1} {4 \ pi \ varepsilon _ {0}}} \ cdot {\ frac {Q \ cdot q} {r ^ {2}}}}$.

The shift from , so that the distance changes from to , corresponds to an electrical work that can be calculated by integrating the counterforce according to the path: ${\ displaystyle q}$${\ displaystyle r_ {1}}$${\ displaystyle r_ {2}}$

${\ displaystyle W_ {12} = \ int _ {r_ {1}} ^ {r_ {2}} F \ cdot \ mathrm {d} r = - {\ frac {Q \ cdot q} {4 \ pi \ varepsilon _ {0}}} \ cdot \ int _ {r_ {1}} ^ {r_ {2}} {\ frac {\ mathrm {d} r} {r ^ {2}}} = {\ frac {Q \ cdot q} {4 \ pi \ varepsilon _ {0}}} \ cdot \ left ({\ frac {1} {r_ {2}}} - {\ frac {1} {r_ {1}}} \ right) }$.

From this formula it is easy to derive the electric potential in the radially symmetrical electric field around the charge , for this purpose the test charge is considered and the infinite distance is chosen as the reference point : ${\ displaystyle Q}$${\ displaystyle q}$

${\ displaystyle \ varphi = \ lim _ {r_ {1} \ to \ infty} {\ frac {W_ {12}} {q}}}$ ;

${\ displaystyle q}$and omitted, after renaming from to results ${\ displaystyle r_ {1}}$${\ displaystyle r_ {2}}$${\ displaystyle r}$

${\ displaystyle \ varphi = {\ frac {Q} {4 \ pi \ varepsilon _ {0} \ cdot r}}}$.

## literature

• Karl Küpfmüller, Wolfgang Mathis, Albrecht Reibiger: Theoretical Electrical Engineering - An Introduction . 19th edition. Springer, 2013, ISBN 978-3-642-37939-0 .