Key Points

There are two types of electric charges, positive and negative. Like charges repel each other, while unlike charges attract each other. The SI unit of charge is the coulomb (C).

Conductors are materials that allow electric charges (electrons) to move freely, such as metals. Insulators are materials that resist the flow of electric charge, like glass, plastic, and wood.

Electric charge is quantised, meaning it exists in discrete packets. The total charge $q$ on a body is always an integral multiple of the basic charge unit $e$, given by the formula $q = ne$, where $e = 1.602 \times 10^{-19} \text{ C}$.

The total charge of an isolated system remains constant; this is the law of conservation of charge. The total charge of a system is the algebraic sum of all individual charges in it, demonstrating the additive nature of charge.

The electrostatic force $F$ between two point charges $q_1$ and $q_2$ separated by a distance $r$ is given by $F = k \frac{|q_1 q_2|}{r^2}$, where $k = \frac{1}{4\pi\epsilon_0} \approx 9 \times 10^9 \text{ N m}^2\text{/C}^2$.

The total force on a given charge due to a number of other charges is the vector sum of the individual forces exerted by each charge. $\mathbf{F}_{total} = \mathbf{F}_{1} + \mathbf{F}_{2} + \mathbf{F}_{3} + \dots$

The electric field $\mathbf{E}$ at a point is the force experienced by a unit positive test charge $q_0$. It is defined as $\mathbf{E} = \frac{\mathbf{F}}{q_0}$ and its SI unit is newtons per coulomb (N/C).

The electric field $\mathbf{E}$ at a distance $r$ from a source charge $q$ is given by $\mathbf{E} = \frac{1}{4\pi\epsilon_0} \frac{q}{r^2} \hat{\mathbf{r}}$. Its direction is radially outward from a positive charge and inward toward a negative charge.

Electric field lines are imaginary curves that represent the electric field. They start from positive charges and end on negative charges, never intersect, and their density indicates the field's strength.

An electric dipole is a pair of equal and opposite charges ($+q$ and $-q$) separated by a distance $2a$. The electric dipole moment $\mathbf{p}$ has a magnitude of $p = q \times 2a$ and is directed from the negative to the positive charge.

For a dipole at large distances $r$, the field on the axial line is $\mathbf{E}_{axial} \approx \frac{2\mathbf{p}}{4\pi\epsilon_0 r^3}$, and on the equatorial plane is $\mathbf{E}_{equatorial} \approx -\frac{\mathbf{p}}{4\pi\epsilon_0 r^3}$.

In a uniform electric field $\mathbf{E}$, a dipole with moment $\mathbf{p}$ experiences a torque given by $\tau = \mathbf{p} \times \mathbf{E}$. The net force on the dipole is zero.

Electric flux $\phi$ is a measure of the flow of the electric field through a given area. For a uniform field $\mathbf{E}$ and a planar area $\mathbf{S}$, the flux is $\phi = \mathbf{E} \cdot \mathbf{S} = ES \cos\theta$.

Gauss's law states that the total electric flux through any closed surface is equal to $\frac{1}{\epsilon_0}$ times the net charge enclosed by the surface. Mathematically, $\phi_{total} = \oint \mathbf{E} \cdot d\mathbf{S} = \frac{q_{enclosed}}{\epsilon_0}$.

Using Gauss's law, the electric field at a perpendicular distance $r$ from an infinitely long straight wire with uniform linear charge density $\lambda$ is $E = \frac{\lambda}{2\pi\epsilon_0 r}$.

The electric field from a uniformly charged infinite plane sheet with surface charge density $\sigma$ is constant and given by $E = \frac{\sigma}{2\epsilon_0}$.

For a spherical shell of radius $R$ with total charge $q$, the electric field outside the shell ($r > R$) is $E = \frac{q}{4\pi\epsilon_0 r^2}$, as if all charge were at the center. The electric field inside the shell ($r < R$) is zero.