# 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

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 .