Principle of Superposition - interaction between any two charges is completely unaffected by presence of other charges.
Not a logical necessity, but an experimental fact (electrostatics)
Coulomb's Law - \(\boldsymbol{F} = \frac{1}{4\pi \epsilon_0} \frac{qQ}{r_{sep}^2} \hat{\boldsymbol{r}}_{sep}\)
Permittivity of free space - \(\epsilon_0 = 8.85 * 10^{-12} \frac{C}{N * m^2}\), where
Separation Vector: \(\overrightarrow{\boldsymbol{r}}_{sep}\ = \boldsymbol{r} - \boldsymbol{r'} \)
\(\boldsymbol{r'}\) - location of q
\(\boldsymbol{r}\) - location of Q
Given several charges \(q_1, q_2, \dotso\) at distances \(r_1, r_2, \dotso\) the Coulomb's force exerted on charge Q:
\(\overrightarrow{\boldsymbol{F}} = \overrightarrow{\boldsymbol{F_1}} + \overrightarrow{\boldsymbol{F_2}} + \dotso =
\frac{1}{4\pi \epsilon_0}(\frac{q_1 Q}{r_{1}^2}\hat{\boldsymbol{r_1}} + \frac{q_2 Q}{r_{2}^2}\hat{\boldsymbol{r_2}} + \dotso) =
\frac{Q}{4\pi \epsilon_0}(\frac{q_1}{r_{1}^2}\hat{\boldsymbol{r_1}} + \frac{q_2}{r_{2}^2}\hat{\boldsymbol{r_2}} + \dotso) \)
\(\overrightarrow{\boldsymbol{F}} = Q \overrightarrow{\boldsymbol{E}} \rightarrow \overrightarrow{\boldsymbol{E}}(\overrightarrow{\boldsymbol{r}}) =
\frac{1}{4\pi \epsilon_0} \sum_{i=1}^{n} \frac{q_i}{r_{i}^2}\hat{\boldsymbol{r_i}}\)
Minimal definition: Intermediate step in calculation of electrical forces (force per unit charge exerted on a charge placed at a given point in space)
Griffith definition: Real physical entity, filling the space in neighborhood of any electric charge.
Can't tell what E-field is, just that it is the reason why Coulomb's force exists.
You have the formula for electric field. Express separation vectors in terms of components.
\(\boldsymbol{E}(\boldsymbol{r}) = \frac{1}{4\pi \epsilon_0} \int \frac{1}{r^2}\hat{\boldsymbol{r}} \partial{q} \text{, where}\ \partial{q} \rightarrow \lambda \partial{l}\ \text{(line charge)} \rightarrow \sigma \partial{a}\ \text{(surface charge)} \rightarrow \rho \partial{\tau}\ \text{(volume charge)} \Rightarrow \\ \boldsymbol{E}(\boldsymbol{r}) = \frac{1}{4\pi \epsilon_0} \int_P \frac{\lambda(\boldsymbol{r'})}{r^2}\hat{\boldsymbol{r}} \partial{l'}\ \text{or}\ \frac{1}{4\pi \epsilon_0} \int_S \frac{\sigma (\boldsymbol{r'})}{r^2}\hat{\boldsymbol{r}} \partial{a'}\ \text{or}\ \require{enclose} \enclose{box}[mathcolor="black"]{\frac{1}{4\pi \epsilon_0} \int_S \frac{\rho (\boldsymbol{r'})}{r^2}\hat{\boldsymbol{r}} \partial{\tau'}}\)
Last expression is the most generic. \(\boldsymbol{r}\) - vector from \(\partial{q}\) to field point from the origin.
Look at Ex 2.2 in the book. Here, you need to break down the dE into two component along principal axies.
Difficult example. Calculate electric field due to side of a square loop and then combine the results from 4 sides.
Write electric field of a smal piece acting at the point distance z away from the center and then integrate ALONG the circular loop
Write electric field of a smal piece acting at the point distance z away from the center and then integrate ALONG the circular loop
Write electric field of a smal surface charge acting at the point distance z away from the center of a sphere.
Distance from the surface charge to that point can be expressed as a function of \(\theta\), R, and z.
Write electric field of a smal surface charge acting at the point distance z away from the center of a sphere.
Distance from the surface charge to that point can be expressed as a function of \(\theta\), R, and z.