2.2A: Point Charge
- Page ID
- 5835
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Let us arbitrarily assign the value zero to the potential at an infinite distance from a point charge \(Q\). “The” potential at a distance \(r\) from this charge is then the work required to move a unit positive charge from infinity to a distance \(r\).
At a distance x from the charge, the field strength is \(\frac{Q}{4\pi\epsilon_0 x^2}\). The work required to move a unit charge from \(x \text{ to }x + δx\) is \(-\frac{Q\,\delta x}{4\pi\epsilon_0 x^2}\). The work required to move unit charge from \(r\) to infinity is \(-\frac{Q}{4\pi\epsilon_0}\int_r^{\infty}\frac{dx}{x^2}=-\frac{Q}{4\pi\epsilon_0 r}\). The work required to move unit charge from infinity to \(r\) is minus this.
Therefore
\[V=+\frac{Q}{4\pi\epsilon_0 r}.\label{2.2.1}\]
The mutual potential energy of two charges \(Q_1 \text{ and }Q_2\) separated by a distance \(r\) is the work required to bring them to this distance apart from an original infinite separation. This is
\[P.E.=+\frac{Q_1Q_2}{4\pi\epsilon_0 r^2}\label{2.2.2}.\]
Before proceeding, a little review is in order.
Field at a distance \(r\) from a charge \(Q\):
\[E=\frac{Q}{4\pi\epsilon_0 r^2},\quad \quad \text{N C}^{-1} \text{ or } \text{V m}^{-1}\]
or, in vector form,
\[\textbf{E}=\frac{Q}{4\pi\epsilon_0 r^2}\hat{\textbf{r}}=\frac{Q}{4\pi\epsilon_0 r^3}\textbf{r}. \quad \quad \text{N C}^{-1}\text{ or }\text{V m}^{-1}\]
Force between two charges, \(Q_1 \text{ and }Q_2\):
\[F=\frac{Q_1Q_2}{4\pi\epsilon r^2}.\quad \quad \text{N}\]
Potential at a distance \(r\) from a charge \(Q\):
\[V=\frac{Q}{4\pi\epsilon_0 r}.\quad \quad \text{V}\]
Mutual potential energy between two charges:
\[\text{P.E.}=\frac{Q_1Q_2}{4\pi\epsilon_0 r}.\quad \quad \text{J}\]
We couldn’t possibly go wrong with any of these, could we?