Electrostatics

Charge ($Q$): Is an inherent property of protons and electrons, which creates a Field ($\vec{E}$) which can in turn interact with the environment.

• Adding electrons yields a negative charge
• Removing electrons yields a positve charge

Field ($\vec{E}$): Is the (infinite) region around anything with charge, which can interact with other things that have charge to create forces. Measured in Newtons per Coulomb ($\frac{N}{C}$)

1. Field vector with one charge source:

$$|E|=k_e\frac{Q}{r^2}$$

Coulomb’s Constant ($k_e$): $8.9875\cdot 10^9\frac{N{m}^2}{C^2}$

Electrostatic Force:

$$\vec{F}=Q_{Object}{\vec{E}}_{Source}$$

Flux ($\varphi$): “total amount” of electric field at any point. Measured in $\frac{N{m}^2}{C}$

\phi_\vec{v} = \int{\vec{V}_{\parallel d\vec{a}}d\vec{a}}

Area vector ($\vec{A},d\vec{a}$): Always normal to the surface of the area.

For a flat surface (where $\vec{A}$ is constant), \phi_\vec{v} = \vec{V}_\parallel A

Gauss’s Law: The total flux on any surface fully enclosing a charged object is proportional to the enclosed objects charge and it quantified by ${\varphi }_{E(enc)}=4\pi k{Q}_{enc}$ $\to$ Field doesn’t “go away”, any time you can enclose a charge, no matter the distance or shape, all of the field is “captured”.

Vacuum Permittivity: ${ϵ}_0=8.854\cdot 10^{-12}$

$$\oint E_{\parallel d\vec{a}}=4\pi {ϵ}_0{Q}_{enc}$$

(integrating over a closed surface area)

For en enclosing spherical shell centered on the charge:

$$E_{\parallel da}{A}_{surf}=4\pi k{Q}_{enc}$$ $$E(r)=\frac{k{Q}_{enc}}{r^2}$$

Linear Charge Density $\lambda$: Coulumb/meter Surface Charge Density $\sigma$ Coulumb/meter

Conducting sheet 4pi k sigma Non conducting sheet 2 pi k sigma

Potential energy stored between charges Electric potential energy from a point charge:

$$V(r)=\frac{k{Q}_{source}}{r}$$

Define one charge to be already in place, since putting it in place took no force, and thus no energy

$$W_{electric}=-\Delta {\mu }_{electric}=-[{\mu }_B={\mu }_A]$$

Electric work = Difference in electric potential energy

$$W_{net}=k_f-k_i=0$$

(there is no kinetic energy at the beginning or end)

$$W_{us}+W_{electric}=0$$ $$W_{\Sigma }=W_{q_1}+W_{q_2}+W_{q_3}+W_{q_4}$$

$W_{q_1}=0$

$$W_{q_n}=-q_2{V}_{12}-V_{\infty }$$

$V_{\infty }$ is always 0

$$W_{\Sigma }=0-q_2(V_{12})-q_3(V_{13}+V_{23})-q_4(V_{14}+V_{24}+V_{34})$$ $$W_{\Sigma }=0-q_2(\frac{k{q}_1}{r_{12}})-q_3(\frac{k{q}_1}{r_{13}}+\frac{k{q}_2}{r_{23}})-q_4(V_{14}+V_{24}+V_{34})$$