Electrical potential

Physical size
Surname	electrical potential
Size type	electrical potential
Formula symbol	${\displaystyle \Phi ,\,\phi ,\,\varphi ,\,V}$
Size and unit system unit dimension SI V M L 2 T −3 I −1 cgs g 1/2 · cm 1/2 · s -1 M 1/2 L 1/2 T −1 Gauss ( cgs ) Statvolt (statV) M 1/2 L 1/2 T −1 HLE ( cgs ) Statvolt (statV) M 1/2 L 1/2 T −1 esE ( cgs ) Statvolt (statV) M 1/2 L 1/2 T −1 emE ( cgs ) Abvolt (abV) M 1/2 L 1/2 T −1 Planck 1 M L 2 T −2 Q −1

The electric potential or electrostatic potential , also electric or electrostatic potential , (Greek small letter Phi ) is a physical quantity in classical electrodynamics . ${\displaystyle \varphi }$

The electrical potential is the quotient of the potential energy of a test charge and the value of this charge : ${\displaystyle q}$

${\displaystyle \varphi ={\frac {E_{\mathrm {pot} }}{q}}}$

A time-invariant, i.e. H. Assuming a static electric field that assigns a potential to every point in space; one therefore speaks of a potential field . The difference between the potentials at two points is called the electrical voltage between these points (see also potential and voltage ).

A potential field can be visualized using equipotential surfaces .

In the SI system of units, the electrical potential has the unit volt ( ) or watt per ampere ( ) or joule per coulomb ( ). ${\displaystyle \mathrm {V} }$ ${\displaystyle \mathrm {W} \,\mathrm {A} ^{-1}}$ ${\displaystyle \mathrm {J} \,\mathrm {C} ^{-1}}$

Electrical potential of a point charge

The electrical potential of a point charge with a charge of different sizes. Blue is negative charge, red is positive.

The electrical potential of a point charge , also called Coulomb potential , is given by in the SI system of units ${\displaystyle q}$

${\displaystyle \varphi ({\vec {x}})={\frac {q}{4\,\pi \,\varepsilon _{0}\,\left|{\vec {x}}\right|}}}$

Here designated

• ${\displaystyle q}$the electric charge
• ${\displaystyle \varepsilon _{0}}$the electric field constant
• ${\displaystyle {\vec {x}}}$ the position of the point under consideration relative to the point charge.

In the Heaviside-Lorentz system of units , due to is simplified ${\displaystyle \varepsilon _{0}=1}$

${\displaystyle \varphi ({\vec {x}})={\frac {q}{4\,\pi \,\left|{\vec {x}}\right|}}}$

Electric potential of a static electric field

In the flame probe experiment , the electrical potential can be measured as a voltage.

If the electric field is known, the potential at the point with the position vector , starting from a zero potential in the position , can be calculated using a curve integral : ${\displaystyle {\vec {E}}}$ ${\displaystyle {\vec {r}}}$ ${\displaystyle {\vec {r}}_{0}}$

${\displaystyle \varphi ({\vec {r}})=-\int _{{\vec {r}}_{0}}^{\vec {r}}{\vec {E}}\cdot \mathrm {d} {\vec {s}}}$

Usually zero potential is chosen. It follows: ${\displaystyle \varphi (\infty )}$

${\displaystyle \varphi ({\vec {r}})=\int _{\vec {r}}^{\infty }{\vec {E}}\cdot \mathrm {d} {\vec {s}}}$

Conversely, the electric field strength can be expressed by the gradient of the potential:

${\displaystyle \Leftrightarrow {\vec {E}}=-{\vec {\nabla }}\varphi \,}$

The Poisson's equation applies to a continuous charge distribution :

${\displaystyle \Delta \varphi =-{\frac {\rho }{\varepsilon _{0}}}}$.

Here designated

• ${\displaystyle \Delta ={\vec {\nabla }}^{2}}$the Laplace operator
• ${\displaystyle \rho }$the charge density
• ${\displaystyle \varepsilon _{0}}$the electric field constant .

Especially for the empty space there is . is therefore a harmonious function . ${\displaystyle \Delta \varphi =0}$ ${\displaystyle \varphi }$

The electrical potential is constant inside a conductor .

Electric potential of a dynamic electric field

The following applies to dynamic electric fields:

${\displaystyle {\vec {\nabla }}\times {\vec {E}}=-{\frac {\partial {\vec {B}}}{\partial t}}\neq 0}$

The electric field can therefore not be represented as a gradient field of the electric potential. Instead, the gradient field of the potential is: ${\displaystyle {\vec {E}}}$

${\displaystyle -{\vec {\nabla }}\varphi ={\vec {E}}+{\frac {\partial {\vec {A}}}{\partial t}}}$

Conversely, the potential at a location can be determined using a curve integral based on a zero potential in any location chosen : ${\displaystyle {\vec {r}}}$ ${\displaystyle {\vec {r}}_{0}}$

${\displaystyle \varphi ({\vec {r}})=-\int _{{\vec {r}}_{0}}^{\vec {r}}\left({\vec {E}}+{\frac {\partial {\vec {A}}}{\partial t}}\right)\cdot \mathrm {d} {\vec {s}}}$

With the usual choice of as zero potential follows: ${\displaystyle \varphi (\infty )}$

${\displaystyle \varphi ({\vec {r}})=\int _{\vec {r}}^{\infty }\left({\vec {E}}+{\frac {\partial {\vec {A}}}{\partial t}}\right)\cdot \mathrm {d} {\vec {s}}}$

Here designated

• ${\displaystyle {\vec {\nabla }}}$the Nabla operator
• ${\displaystyle {\vec {B}}}$the magnetic flux density
• ${\displaystyle {\vec {A}}}$the magnetic vector potential

With the Lorenz calibration , the Poisson equation follows for a continuous charge distribution : ${\displaystyle {\vec {\nabla }}{\vec {A}}=-{\frac {1}{c^{2}}}{\frac {\partial \varphi }{\partial t}}}$

${\displaystyle \Delta \varphi -{\frac {1}{c^{2}}}{\frac {\partial ^{2}\varphi }{\partial t^{2}}}=-{\frac {\rho }{\varepsilon _{0}}}}$

Here designated

• ${\displaystyle \Delta ={\vec {\nabla }}^{2}}$the Laplace operator
• ${\displaystyle \rho }$the charge density
• ${\displaystyle \varepsilon _{0}}$the electric field constant
• ${\displaystyle c}$the speed of light .

For stationary fields we have and , so that the formulas change back to those for static fields. ${\displaystyle {\frac {\partial {\vec {A}}}{\partial t}}=0}$ ${\displaystyle {\frac {\partial \varphi }{\partial t}}=0}$

Individual evidence

1. ↑ ^{a b} Wolfgang Demtröder: Experimentalphysik 2 Electricity and Optics . 7., corr. and exp. Edition. Springer-Verlag GmbH, Berlin 2018, ISBN 978-3-662-55789-1 .