Summary

Coulomb’s Law: Force between two point charges $q_1$ and $q_2$ separated by distance $r$ in vacuum is $F = \frac{1}{4\pi\varepsilon_0} \frac{q_1 q_2}{r^2}$, where $\frac{1}{4\pi\varepsilon_0} = 9 \times 10^9 \, \text{N m}^2/\text{C}^2$. Force is along the line joining charges, attractive for opposite signs, repulsive for like signs.

Electric Field: Field at a point is force per unit positive test charge: $\vec{E} = \frac{\vec{F}}{q_0}$. For a point charge $q$, $\vec{E} = \frac{1}{4\pi\varepsilon_0} \frac{q}{r^2} \hat{r}$.

Superposition Principle: Net force or field due to multiple charges is the vector sum of individual contributions.

Electric Field Lines: Start at positive charges, end at negative charges; never cross; tangent gives field direction; density indicates field strength.

Electric Dipole: Two equal and opposite charges $+q$ and $-q$ separated by distance $2a$. Dipole moment $\vec{p} = q \cdot 2a \hat{p}$ (from $-q$ to $+q$).

Field of a Dipole:

• On axial line: $E_{\text{axial}} = \frac{1}{4\pi\varepsilon_0} \frac{2p}{r^3}$ (for $r \gg a$).
• On equatorial line: $E_{\text{equatorial}} = \frac{1}{4\pi\varepsilon_0} \frac{p}{r^3}$ (for $r \gg a$).

Torque on a Dipole in Uniform Field: $\vec{\tau} = \vec{p} \times \vec{E}$, magnitude $\tau = pE \sin\theta$. Potential energy $U = -\vec{p} \cdot \vec{E}$.

Gauss’s Law: Total electric flux through a closed surface equals $\frac{1}{\varepsilon_0}$ times the net charge enclosed: $\oint \vec{E} \cdot d\vec{S} = \frac{q_{\text{enc}}}{\varepsilon_0}$.

Applications of Gauss’s Law:

• Field due to an infinite line charge: $E = \frac{\lambda}{2\pi\varepsilon_0 r}$.
• Field due to an infinite plane sheet: $E = \frac{\sigma}{2\varepsilon_0}$ (independent of distance). …