16.5: Direct Calculation of Electrical Quantities from Charge Distributions (Summary)
Key Terms
| continuous charge distribution | total source charge composed of so large a number of elementary charges that it must be treated as continuous, rather than discrete |
| infinite straight wire | straight wire whose length is much, much greater than either of its other dimensions, and also much, much greater than the distance at which the field is to be calculated |
| linear charge density | amount of charge in an element of a charge distribution that is essentially one-dimensional (the width and height are much, much smaller than its length); its units are C/m |
| surface charge density | amount of charge in an element of a two-dimensional charge distribution (the thickness is small); its units are \(\displaystyle C/m^2\) |
| volume charge density | amount of charge in an element of a three-dimensional charge distribution; its units are \(\displaystyle C/m^3\) |
Key Equations
| Coulomb’s law | \(\displaystyle \vec{F_{12}}(r)=\frac{1}{4πε_0}\frac{q_1q_2}{r^2_{12}}\hat{r_{12}}\) |
| Superposition of electric forces | \(\displaystyle \vec{F}(r)=\frac{1}{4πε_0}Q \sum_{i=1}^N\frac{q_i}{r^2_i}\hat{r_i}\) |
| Electric force due to an electric field | \(\displaystyle \vec{F}=Q\vec{E}\) |
| Electric field at point P | \(\displaystyle \vec{E}(P)≡\frac{1}{4πε_0}\sum_{i=1}^N\frac{q_i}{r^2_i}\hat{r_i}\) |
| Field of an infinite wire | \(\displaystyle \vec{E}(z)=\frac{1}{4πε_0}\frac{2λ}{z}\hat{k}\) |
| Field of an infinite plane | \(\displaystyle \vec{E}=\frac{σ}{2ε_0}\hat{k}\) |
| Dipole moment | \(\displaystyle \vec{p}≡q\vec{d}\) |
| Torque on dipole in external E-field | \(\displaystyle \vec{τ}=\vec{p}×\vec{E}\) |
| Electric field of a continuous charge distribution | \(\displaystyle \vec{E}=k∫\frac{dq \hat{r}}{r}\) |
| Electric potential of a continuous charge distribution | \(\displaystyle V_P=k∫\frac{dq}{r}\) |
Summary
Calculating Electric Fields of Charge Distributions
-
A very large number of charges can be treated as a continuous charge distribution, where the calculation of the field requires integration. Common cases are:
- one-dimensional (like a wire); uses a line charge density \(\displaystyle λ\)
- two-dimensional (metal plate); uses surface charge density \(\displaystyle σ\)
- three-dimensional (metal sphere); uses volume charge density \(\displaystyle ρ\)
- The “source charge” is a differential amount of charge dq. Calculating dq depends on the type of source charge distribution:
\(\displaystyle dq=λdl;dq=σdA;dq=ρdV\).
- The field of continuous charge distributions may be calculated with \(\displaystyle \vec{E}=k∫\frac{dq \hat{r}}{r}\).
- Symmetry of the charge distribution is usually key.
- Important special cases are the field of an “infinite” wire and the field of an “infinite” plane.
Electric Dipoles
- If a permanent dipole is placed in an external electric field, it results in a torque that aligns it with the external field.
- If a nonpolar atom (or molecule) is placed in an external field, it gains an induced dipole that is aligned with the external field.
- The net field is the vector sum of the external field plus the field of the dipole (physical or induced).
- The strength of the polarization is described by the dipole moment of the dipole, \(\displaystyle \vec{p}=q\vec{d}\).
Calculating Electric Potential of Charge Distributions
- The potential of continuous charge distributions may be calculated with \(\displaystyle V_P=k∫\frac{dq}{r}\).
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)
.