3.8: Electric Potential (Summary)
- Page ID
- 100340
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Key Terms
| Term | Definition |
|---|---|
| electric dipole | system of two equal but opposite charges a fixed distance apart |
| 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 |
| 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 |
Key Equations
| Description | Equation |
|---|---|
| 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 | \(\displaystyle V=\frac{U}{q}=−∫^P_R\vec{E}⋅\vec{dl}\) |
| 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 potential of a continuous charge distribution | \(\displaystyle V_P=k∫\frac{dq}{r}\) |
Summary
Work and Energy
- The work done by a force, acting over a finite path, is the integral of the infinitesimal increments of work done along the path, which are given by the dot product of the force and the infinitesimal displacements.
- The kinetic energy of a particle is the product of one-half its mass and the square of its speed (for non-relativistic speeds), and the kinetic energy of a system is the sum of the kinetic energies of all the particles in the system.
- The integral for the net work done on the particle is equal to the change in the particle’s kinetic energy. This is the work-kinetic energy theorem.
- For a single-particle system, the difference of potential energy is the opposite of the work done by the forces acting on the particle as it moves from one position to another.
- A conservative force is one for which the work done is independent of path. Equivalently, a force is conservative if the work done over any closed path is zero.
- If non-conservative forces do no work and there are no external forces, the mechanical energy of a particle stays constant. This is a statement of the conservation of mechanical energy and there is no change in the total mechanical energy.
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.
- 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.
Electric Potential Energy of Point Charges
- 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.
Electric Potential
- 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,
\(\displaystyle 1\,\mathrm{eV}=(1.60×10^{−19}\,\mathrm{C})(1\,\mathrm{V})=(1.60×10^{−19}\,\mathrm{C})(1\,\mathrm{J}/\mathrm{C})=1.60×10^{−19}\,\mathrm{J}\).
Electric Potential of a Point Charge
- Electric potential is a scalar, whereas electric field is a vector.
- The 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 \(\displaystyle \vec{p}=q\vec{d}\).
Common Models of Electric Potential
- Continuous charge distributions may be calculated with \(\displaystyle V_P=k∫\frac{dq}{r}\).
- Results are for the electric potential provided for common continuous charge distributions including a line segment, ring, disk, and infinite line.
Contributors and Attributions
Samuel J. Ling (Truman State University), Jeff Sanny (Loyola Marymount University), and Bill Moebs with many contributing authors. This work is licensed by OpenStax University Physics under a Creative Commons Attribution License (by 4.0).