# Potential energy

Hydroelectric power plants use the potential energy of a reservoir . The greater the amount of water stored and the greater the height difference of the barrage, the more electrical energy the power plant can deliver.

The potential energy (also called potential or positional energy ) describes the energy of a body in a physical system , which is determined by its position in a force field or by its current (mechanical) configuration.

For example, in a gravitational field, the “potential energy” is the energy that a body has due to its altitude : If a stone falls from a height of 20 meters, it has twice as much work capacity as if it fell 10 meters. During the fall, the potential energy is converted into kinetic energy or other forms of energy and diminishes. In hydropower plants, the potential energy of the water in a reservoir can be converted into electrical energy.

Like other forms of energy, potential energy is a state variable of a physical system. In a closed system , the potential energy can increase or decrease when the state changes, for example when the body is moved, when it changes in height or when an atom is excited by radiation. But then another form of energy (e.g. kinetic energy, electrical field energy ) always decreases or increases to the same extent. This fact of experience is expressed by the law of conservation of energy .

The SI unit of potential energy is the joule ( symbol J). E pot or U is used as a symbol for the potential energy, V is common in theoretical physics . The term potential is often spoken of inaccurately when the potential energy is meant.

## Potential energy in the gravitational field

### introduction

As an introduction, let's consider a cyclist who travels a flat stretch, then goes up a mountain and down last. The observation should initially take place without frictional forces .

On a level route, the cyclist travels at a certain speed, which corresponds to a certain kinetic energy. If he drives up the mountain, he has to expend more energy in order to maintain the speed (and thus kinetic energy). Due to the conservation of energy, however, no energy can be lost and the energy that the cyclist expends more on the climb has to flow somewhere: The more expended energy is converted into potential energy . The higher it rises, the more potential energy the cyclist has. On the other hand, when descending, the cyclist has to brake in order to maintain his speed and thus keep his kinetic energy constant. If he doesn't brake, he gets faster and has more and more kinetic energy. However, due to the conservation of energy, the increase in its kinetic energy cannot go hand in hand without the loss of another form of energy. The increase in kinetic energy is equal to the loss of potential energy.

#### Cyclists in detail with friction

Representation of the forces on the cyclist. The weight F G , which can be broken down into the downhill force F GH and normal force F GN , and the frictional force F R are shown

The cyclist can easily reach 20 km / h on a level route, since he only has to compete against air resistance and rolling friction . If he now comes to an ascending stretch of the road , he has to exert himself more than before at the same speed. After reaching the crest, it goes downhill and the cyclist rolls on without pedaling, even having to brake so that he doesn't get too fast.

Two forces act on the rider and the bike : the frictional force and the weight force . In the first section of the route, the weight force is perpendicular to the road and therefore does not have any force components in the direction of movement after the force decomposition has been applied. If there is an increase, the decomposition of the weight results in a force component against the direction of movement. After crossing the summit, gravity has a component in the direction of movement and against the frictional force.

For a movement against the force of weight, work must be expended on the body , which is now stored in it as potential energy. With a movement that contains a component in the direction of weight, the body does work and its potential energy decreases. The path component in the direction of weight force is called height and together with the force results:

${\ displaystyle E _ {\ mathrm {pot}} \ equiv U _ {\ mathrm {G}} = F _ {\ mathrm {G}} \, h = m \, g \, h}$
• ${\ displaystyle m}$- mass
• ${\ displaystyle g}$- gravitational acceleration
• ${\ displaystyle h}$- Height above the ground (significantly less than the earth's radius ; )${\ displaystyle h \ ll R _ {\ mathrm {E}}}$

### More general description

In general, the gravitational field strength and thus the weight is location-dependent. The following applies:

{\ displaystyle {\ begin {aligned} U _ {\ mathrm {G}} ({\ vec {r}}) & = - \ int _ {{\ vec {r}} _ {0}} ^ {\ vec { r}} {\ vec {F}} ({\ vec {r}} ^ {\ prime}) \ cdot d {\ vec {r}} ^ {\ prime} = - \ int _ {{\ vec {r }} _ {0}} ^ {\ vec {r}} m \, {\ vec {g}} ({\ vec {r}} ^ {\ prime}) \ cdot \ mathrm {d} {\ vec { r}} ^ {\ prime} \ end {aligned}}}

The negative sign results from the fact that one has to move something against the acting force in order to increase the potential energy. describes the reference point at which the potential energy of a particle should disappear. It replaces the integration constant that otherwise occurs during integration. Usually one chooses the earth's surface (see first example) or infinity (see second example). The upper limit of the integral corresponds to the position of the considered particle, not to be confused with the integration variable . ${\ displaystyle {\ vec {r_ {0}}}}$${\ displaystyle {\ vec {r}}}$${\ displaystyle {\ vec {r ^ {\ prime}}}}$

#### Example: Potential energy on the earth's surface

If one sets constant (which applies approximately to small height differences on the earth's surface), the equation described in the previous section results again: ${\ displaystyle {\ vec {g}} ({\ vec {r}})}$

{\ displaystyle {\ begin {aligned} U _ {\ mathrm {G}} ({\ vec {r}}) & = - \ int _ {R} ^ {R + h} m \, {\ vec {g} } \ cdot \ mathrm {d} {\ vec {r}} = mgh \ end {aligned}}}

#### Example: Potential energy on a planet's surface

Potential energy W pot and gravitational potential V in the vicinity of a central
mass

If one considers a system of a planet and a sample particle far away from the planet's surface, the above approximation is no longer sufficient; the local gravitational field strength varies with the distance from the center of mass of the planet. A more precise description is possible using Newton's law of gravitation ,

${\ displaystyle {\ vec {F _ {\ mathrm {G}}}} ({\ vec {r}}) = - {\ frac {GMm} {| {\ vec {r}} | ^ {3}}} {\ vec {r}} = - {\ frac {GMm} {r ^ {2}}} {\ vec {e_ {r}}}; ​​\ quad r = | {\ vec {r}} |; \ quad {\ vec {e_ {r}}} = {\ frac {\ vec {r}} {r}}}$.

In this type of observation one often chooses the reference point at the infinitely distant, i.e. H. . With this choice, the potential energy can only assume negative values. The potential energy of the particle on the planet's surface then corresponds to the work that has to be done to transport this particle to infinity, i.e. to remove it from the gravitational field. The potential energy of the particle is minimal on the surface of the planet and maximal at infinity. With the agreement that the origin is in the center of the planet, the planet has a radius and , one obtains ${\ displaystyle U _ {\ mathrm {G}} (r = \ infty) = 0}$${\ displaystyle R}$${\ displaystyle \ mathrm {d} {\ vec {r}} \ cdot {\ vec {e_ {r}}} = \ mathrm {d} r}$

${\ displaystyle U _ {\ mathrm {min}} = - \ int _ {\ infty} ^ {R} {\ vec {F _ {\ mathrm {G}}}} ({\ vec {r}}) \ cdot \ mathrm {d} {\ vec {r}} = - \ int _ {\ infty} ^ {R} - {\ frac {GMm} {r ^ {2}}} \, \ mathrm {d} r = GMm \ int _ {\ infty} ^ {R} {\ frac {1} {r ^ {2}}} \, \ mathrm {d} r = GMm \ left [- {\ frac {1} {r}} \ right ] _ {\ infty} ^ {R} = - {\ frac {GMm} {R}} = - m \, g \, R}$

as potential energy of the sample particle on the planet's surface. The last step was the new planet-dependent constant

${\ displaystyle g = {\ frac {GM} {R ^ {2}}}}$

Are defined. Here is the mass of the planet, the mass of the sample particle and the gravitational constant . ${\ displaystyle M}$${\ displaystyle m}$${\ displaystyle G}$

## Potential energy of a tensioned spring

From the spring force

${\ displaystyle F (x) = - kx}$,

results for the potential energy

${\ displaystyle U (x) = - \ int _ {0} ^ {x} F (x ^ {\ prime}) \ mathrm {d} x ^ {\ prime} = {1 \ over 2} kx ^ {2 }}$.

Here is the spring constant and the deflection of the spring from the rest position. ${\ displaystyle k}$${\ displaystyle x}$

## Potential energy and the law of conservation of energy

In a closed system without energy exchange with the environment and neglecting any friction , the energy conservation law of classical mechanics applies at all times :

${\ displaystyle E = T + U = {\ text {const.}}}$
• ${\ displaystyle U}$ - potential energy
• ${\ displaystyle T}$- kinetic energy
• ${\ displaystyle E}$- mechanical energy

In words: The sum of potential and kinetic energy, including the rotational energy , is constant and corresponds to the total energy of the mechanical system.

In the Hamilton formalism this equation is called

${\ displaystyle H = \ sum _ {k} p_ {k} {\ dot {q}} _ {k} -L = T + U}$

written, where is the Hamilton function and the Lagrange function . ${\ displaystyle H}$${\ displaystyle L}$

## Formal definition

Potential and gradient fields in physics;
Scalar fields (potential fields):
V G - gravitational potential
E pot - potential energy
V C - Coulomb potential
Vector fields (gradient fields):
g - gravitational acceleration
F - force
E - electrical field strength
Electric field: equipotential lines (red) and field lines (black) for two point-like concentrated charges

Since a conservative force field defines the force on a specimen at any location and is mathematically a gradient field , there is a scalar field equivalent to the force field . This is the potential energy for the respective location. It follows from the inversion of the work integral that an increase in energy along a path requires a force component in the opposite direction of the path. By decomposing the force field into Cartesian components , the following partial derivatives result depending on the location : ${\ displaystyle {\ vec {F}} ({\ vec {r}})}$ ${\ displaystyle U ({\ vec {r}})}$

${\ displaystyle - {\ vec {F}} ({\ vec {r}}) = {\ frac {\ partial U ({\ vec {r}})} {\ partial x}} {\ vec {e} } _ {x} + {\ frac {\ partial U ({\ vec {r}})} {\ partial y}} {\ vec {e}} _ {y} + {\ frac {\ partial U ({ \ vec {r}})} {\ partial z}} {\ vec {e}} _ {z}}$

In general, this can be expressed using the Nabla operator . ${\ displaystyle {\ vec {\ nabla}}}$

${\ displaystyle {\ vec {F}} ({\ vec {r}}) = - {\ vec {\ nabla}} U ({\ vec {r}})}$

The reversal of the derivative leads to the integral and determines the change in potential energy in the force field as a work integral with a negative sign. This also shows the transferability to different force fields.

${\ displaystyle U ({\ vec {r}}) = - \ int \ mathrm {d} W ({\ vec {r}}) = - \ int {\ vec {F}} ({\ vec {r} }) \ cdot \ mathrm {d} {\ vec {r}}}$

In order to increase the potential energy of a body, field work must be done against the forces of a conservative force field. Every body with mass in a gravitational field has potential energy. However, this can only be increased or decreased if the body is shifted against or in the direction of the gravitational force. With a displacement perpendicular to the field lines , the body retains its potential energy. Such an area is called an equipotential surface or line and corresponds to a contour line on the map. The field line, on the other hand, describes the direction of the slope.

Path independence of the displacement work in the conservative force field

As long as there are no frictional losses or other interactions with the environment, the principle of path independence applies to a shift in conservative force fields. This means that regardless of the path taken, the same amount of field work has to be done so that a body can get from the starting point to the destination. Here the law of conservation of energy plays itself again, since the work corresponds to the change in energy.

The choice of the reference level can be made arbitrarily, but pragmatic reasons reduce the selection. In case of doubt , the starting point of the examined body is always suitable as a zero level . In the case of a gravitational field, the earth's surface is often the reference point or generally the lowest point in the environment. In addition, the reference point can be moved to an infinitely distant location ( ). The reverse of this forms the maximum potential energy at which a body is moved from its starting point out of the force field, assuming a central force field. ${\ displaystyle | {\ vec {r}} | \ rightarrow \ infty}$

${\ displaystyle U _ {\ mathrm {max}} = - \ int _ {R} ^ {\ infty} {\ vec {F}} (r) \ mathrm {d} r}$

With electrical charges of the same sign, this leads to minimal potential energy.

### Example: Potential energy in an electric field

The force on a charge in a given electric field is calculated from:

${\ displaystyle {\ vec {F}} _ {\ mathrm {C}} ({\ vec {r}}) = q \, {\ vec {E}} ({\ vec {r}}).}$

Inserting it into the working integral shows the relationship between the potential energy of a charge and the Coulomb potential, which also represents a scalar field. The only difference between the two fields is the proportionality factor charge:

{\ displaystyle {\ begin {aligned} U _ {\ mathrm {C}} ({\ vec {r}}) & = - \ int q \, {\ vec {E}} ({\ vec {r}}) \ cdot \ mathrm {d} {\ vec {r}} \\\, & = q \, \ phi ({\ vec {r}}). \ end {aligned}}}

${\ displaystyle \ phi}$is the so-called Coulomb potential .

### Relationship between potential energy and potential

The concept of potential energy is closely related to the concept of potential, which is an equivalent representation of a conservative force field . The potential energy of a physical system is the product of the coupling constant of the particle with respect to the force field to which it is exposed (e.g. mass in the case of the gravitational field , charge in the case of the electric field ) and the potential of the force field: ${\ displaystyle E _ {\ mathrm {pot}}}$ ${\ displaystyle k}$${\ displaystyle {\ vec {F}} ({\ vec {r}})}$${\ displaystyle m}$${\ displaystyle q}$${\ displaystyle \ phi ({\ vec {r}})}$

${\ displaystyle E _ {\ mathrm {pot}} ({\ vec {r}}): = k \, \ phi ({\ vec {r}}).}$

The definition is related to the force field. Based on this definition, the potential energy is only defined for particles in conservative force fields and the zero point of the energy scale can be determined as desired. ${\ displaystyle {\ vec {F}} ({\ vec {r}}) = - {\ vec {\ nabla}} E _ {\ mathrm {pot}} ({\ vec {r}})}$

#### Example gravitational field

The force on a specimen of the mass in a given gravitational field is calculated from: ${\ displaystyle m}$

${\ displaystyle {\ vec {F}} _ {\ mathrm {G}} ({\ vec {r}}) = m \, {\ vec {g}} ({\ vec {r}}).}$

By inserting it into the work integral, the relationship between the potential energy of a mass and the gravitational potential is shown , which is also a scalar field. ${\ displaystyle \ phi _ {\ mathrm {G}}}$

{\ displaystyle {\ begin {aligned} U _ {\ mathrm {G}} ({\ vec {r}}) & = - \ int {\ vec {F}} d {\ vec {r}} = - m \ int {\ vec {g}} ({\ vec {r}}) \ cdot \ mathrm {d} {\ vec {r}} \\\, & = m \, \ phi _ {\ mathrm {G}} ({\ vec {r}}). \ end {aligned}}}

The factor clearly describes the dependence on the test specimen and the potential describes the field property. ${\ displaystyle m}$${\ displaystyle \ phi _ {\ mathrm {G}} ({\ vec {r}})}$

## Individual evidence

1. ^ Alonso, Finn : Physics, Addison-Wesley (1992), ISBN 0-201-56518-8 , p. 169.
2. Demtröder : Experimentalphysik 1, Springer (2008), ISBN 978-3-540-79294-9 , p. 63.
3. Gerthsen: Physics (21st edition); P. 49

## Remarks

1. z. B. the configuration of a spring : A "tensioned" spring has more potential energy than a "relaxed" spring.