# 7.S: Electric Potential (Summary)

## Key Terms

electric dipole |
system of two equal but opposite charges a fixed distance apart |

electric dipole moment |
quantity defined as \(\displaystyle \{vec{p}=q\vec{d}\) for all dipoles, where the vector points from the negative to positive charge |

electric potential |
potential energy per unit charge |

electric potential difference |
the change in potential energy of a charge q moved between two points, divided by the charge. |

electric potential energy |
potential energy stored in a system of charged objects due to the charges |

electron-volt |
energy given to a fundamental charge accelerated through a potential difference of one volt |

electrostatic precipitators |
filters that apply charges to particles in the air, then attract those charges to a filter, removing them from the airstream |

equipotential line |
two-dimensional representation of an equipotential surface |

equipotential surface |
surface (usually in three dimensions) on which all points are at the same potential |

grounding |
process of attaching a conductor to the earth to ensure that there is no potential difference between it and Earth |

ink jet printer |
small ink droplets sprayed with an electric charge are controlled by electrostatic plates to create images on paper |

photoconductor |
substance that is an insulator until it is exposed to light, when it becomes a conductor |

Van de Graaff generator |
machine that produces a large amount of excess charge, used for experiments with high voltage |

voltage |
change in potential energy of a charge moved from one point to another, divided by the charge; units of potential difference are joules per coulomb, known as volt |

xerography |
dry copying process based on electrostatics |

## Key Equations

Potential energy of a two-charge system | \(\displaystyle U(r)=k\frac{qQ}{r}\) |

Work done to assemble a system of charges | \(\displaystyle W_{12⋯N}=\frac{k}{2}\sum_i^N\sum_j^N\frac{q_iq_j}{r_{ij}}\) for \(\displaystyle i≠j\) |

Potential difference | \(\displaystyle ΔV=\frac{ΔU}{q}\) or \(\displaystyle ΔU=qΔV\) |

Electric potential | \(\displaystyle V=\frac{U}{q}=−∫^P_R\vec{E}⋅\vec{dl}\) |

Potential difference between two points | \(\displaystyle ΔV_{AB}=V_B−V_A=−∫^B_A\vec{E}⋅\vec{dl}\) |

Electric potential of a point charge | \(\displaystyle V=\frac{kq}{r}\) |

Electric potential of a system of point charges | \(\displaystyle V_P=k\sum_1^N\frac{q_i}{r_i}\) |

Electric dipole moment | \(\displaystyle \vec{p}=q\vec{d}\) |

Electric potential due to a dipole | \(\displaystyle V_P=k\frac{\vec{p}⋅\hat{r}}{r^2}\) |

Electric potential of a continuous charge distribution | \(\displaystyle V_P=k∫\frac{dq}{r}\) |

Electric field components | \(\displaystyle E_x=−\frac{∂V}{∂x},E_y=−\frac{∂V}{∂y},E_z=−\frac{∂V}{∂z}\) |

Del operator in Cartesian coordinates | \(\displaystyle \vec{∇}=\hat{i}\frac{∂}{∂x}+\hat{j}\frac{∂}{∂y}+\hat{k}\frac{∂}{∂z}\) |

Electric field as gradient of potential | \(\displaystyle \vec{E}=−\vec{∇}V\) |

Del operator in cylindrical coordinates | \(\displaystyle \vec{∇}=\hat{r}\frac{∂}{∂r}+\hat{φ}\frac{1}{r}\frac{∂}{∂φ}+\hat{z}\frac{∂}{∂z}\) |

Del operator in spherical coordinates | \(\displaystyle \vec{∇}=\hat{r}\frac{∂}{∂r}+\hat{θ}\frac{1}{r}\frac{∂}{∂θ}+\hat{φ}\frac{1}{rsinθ}\frac{∂}{∂φ}\) |

## Summary

#### 7.1 Electric Potential Energy

- The work done to move a charge from point
*A*to*B*in an electric field is path independent, and the work around a closed path is zero. Therefore, the electric field and electric force are conservative. - We can define an electric potential energy, which between point charges is \(\displaystyle U(r)=k\frac{qQ}{r}\), with the zero reference taken to be at infinity.
- The superposition principle holds for electric potential energy; the potential energy of a system of multiple charges is the sum of the potential energies of the individual pairs.

#### 7.2 Electric Potential and Potential Difference

- Electric potential is potential energy per unit charge.
- The potential difference between points
*A*and*B*, \(\displaystyle V_B−V_A\), that is, the change in potential of a charge q moved from*A*to*B*, is equal to the change in potential energy divided by the charge. - Potential difference is commonly called voltage, represented by the symbol \(\displaystyle ΔV\):

\(\displaystyle ΔV=\frac{ΔU}{q}\) or \(\displaystyle ΔU=qΔV.\)

- An electron-volt is the energy given to a fundamental charge accelerated through a potential difference of 1 V. In equation form,

\(\displaystyle 1eV=(1.60×10^{−19}C)(1V)=(1.60×10^{−19}C)(1J/C)=1.60×10^{−19}J\)..

#### 7.3 Calculations of Electric Potential

- Electric potential is a scalar whereas electric field is a vector.
- Addition of voltages as numbers gives the voltage due to a combination of point charges, allowing us to use the principle of superposition: \(\displaystyle V_P=k\sum_1^N\frac{q_i}{r_i}|).
- An electric dipole consists of two equal and opposite charges a fixed distance apart, with a dipole moment \(\displaystyle \vec{p}=q\vec{d}\).
- Continuous charge distributions may be calculated with \(\displaystyle V_P=k∫\frac{dq}{r}\).

#### 7.4 Determining Field from Potential

- Just as we may integrate over the electric field to calculate the potential, we may take the derivative of the potential to calculate the electric field.
- This may be done for individual components of the electric field, or we may calculate the entire electric field vector with the gradient operator.

#### 7.5 Equipotential Surfaces and Conductors

- An equipotential surface is the collection of points in space that are all at the same potential. Equipotential lines are the two-dimensional representation of equipotential surfaces.
- Equipotential surfaces are always perpendicular to electric field lines.
- Conductors in static equilibrium are equipotential surfaces.
- Topographic maps may be thought of as showing gravitational equipotential lines.

#### 7.6 Applications of Electrostatics

- Electrostatics is the study of electric fields in static equilibrium.
- In addition to research using equipment such as a Van de Graaff generator, many practical applications of electrostatics exist, including photocopiers, laser printers, ink jet printers, and electrostatic air filters.

## Contributors

Paul Peter Urone (Professor Emeritus at California State University, Sacramento) and Roger Hinrichs (State University of New York, College at Oswego) with Contributing Authors: Kim Dirks (University of Auckland) and Manjula Sharma (University of Sydney). This work is licensed by OpenStax University Physics under a Creative Commons Attribution License (by 4.0).