2.2: Potential Near Various Charged Bodies
- Page ID
- 5418
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The geometry of the system has a strong effect on the electric potential. Several geometries are discussed below.
- 2.2A: Point Charge
- 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.
- 2.2B: Spherical Charge Distributions
- Outside any spherically-symmetric charge distribution, the field is the same as if all the charge were concentrated at a point in the center, and so, then, is the potential.
- 2.2C: Long Charged Rod
- This page explains the electric field and potential difference around a long charged rod, with the electric field at distance \(r\) represented by \(\frac{\lambda}{2\pi\epsilon_0 r}\). It derives the potential difference between points at distances \(a\) and \(b\) from the rod, resulting in the equation \(V_a - V_b = \frac{\lambda}{2\pi\epsilon_0}\ln(b/a)\), emphasizing the logarithmic relationship between potential difference and distance.
- 2.2D: Large Plane Charged Sheet
- This page explains the electric field from a large charged sheet, represented as \(\frac{\sigma}{2\epsilon_0}\). It also derives the potential difference, \(V_a - V_b\), between two points at distances \(a\) and \(b\) from the sheet, demonstrating that the potential difference is directly proportional to the charge density and the distance between the points.
- 2.2E: Potential on the Axis of a Charged Ring
- This page explains the electric potential on the axis of a charged ring, referencing section 1.6.4. It presents a formula for potential derived from the electric field and conceptual understanding, expressed as \( V=\frac{Q}{4\pi\epsilon_0 (a^2+x^2)^{1/2}} \). This formula underscores the relationship between charge, distance, and electrostatic potential.
- 2.2F: Potential in the Plane of a Charged Ring
- This page explores calculating electric potential at a point in the plane of a charged ring by deriving the potential from a charge element and integrating for the total potential. It highlights numerical integration methods, emphasizing Gaussian quadrature for accuracy due to the complexity of analytic solutions. Two series expansions for potential are introduced, with Series II showing quicker convergence via Legendre polynomials.
- 2.2G: Potential on the Axis of a Charged Disc
- This page focuses on deriving the electric potential on the axis of a charged disc, referencing section 1.6.5 for related details on the electric field. It presents the potential formula \(V=\frac{2Q}{4\pi\epsilon_0 a^2}[(a^2+x^2)^{1/2}-x]\) and encourages readers to engage with the derivation for a better grasp of electrostatic concepts.