Electric Charges and Fields — Notes

Electric charge is a basic property of matter that makes it exert and feel electric forces. SI unit: coulomb (C).

• Two kinds: positive and negative. Like charges repel; unlike charges attract. (Glass rubbed with silk becomes positive; the silk negative.)
• Additivity: the total charge of a system is the algebraic sum of its charges.
• Conservation: the net charge of an isolated system never changes. Charging by rubbing only transfers electrons.
• Quantisation: $q = n e$, where $n$ is an integer and $e = 1.6 \times 10^{-19}$ C. Everyday charges contain so many $e$ that charge looks continuous.

Conductors (metals, the human body, tap water) have free charge carriers; insulators (glass, plastic, dry wood) hold charge where it is placed.

Charging by induction: bring a charged rod near an insulated conductor; charges separate. Earth the far side, remove the earth connection, then remove the rod. The conductor keeps a charge opposite to the rod's. A charged body attracts a neutral light object (a paper bit) because it induces opposite charge on the near side.

For two point charges a distance $r$ apart in vacuum:

Vector form: $\vec F_{21} = k\dfrac{q_1 q_2}{r^2}\hat r_{12}$ (force on $q_2$; $\hat r_{12}$ points from $q_1$ to $q_2$), and $\vec F_{12} = -\vec F_{21}$.

In a medium of dielectric constant $\varepsilon_r$: $F = \dfrac{F_\text{vacuum}}{\varepsilon_r}$.

Superposition: the force on a charge due to many charges is the vector sum of the individual Coulomb forces, each calculated as if the others were absent.

Method for multi-charge problems: (1) draw each force with its direction from the signs; (2) find magnitudes; (3) add by components or, for perpendicular forces, by Pythagoras; (4) state magnitude and direction.

Worked result (deck example 2): forces of 0.40 N and 0.30 N at right angles give 0.50 N.

The electric field at a point is the force per unit small positive test charge: $\vec E = \vec F / q_0$ (unit N C$^{-1}$). A charge $q$ placed there feels $\vec F = q\vec E$ (opposite to $\vec E$ if $q < 0$).

Point charge: $E = k\dfrac{|q|}{r^2}$, radially away from a positive charge and towards a negative one.

System of charges: $\vec E = \sum \vec E_i$ (superposition).

Field lines — properties:

• tangent gives the direction of $\vec E$; start on $+$ charges, end on $-$ charges (or at infinity);
• never cross; never form closed loops in electrostatics;
• crowding shows strength; parallel equally spaced lines show a uniform field;
• meet a conductor's surface at right angles.

Neutral point: where the resultant field is zero. Between two like charges it lies between them (nearer the smaller charge); for unlike charges it lies outside, on the side of the smaller charge.

Dipole: charges $+q$ and $-q$ separated by $2a$. Dipole moment $\vec p = q(2a)$, from $-q$ to $+q$, unit C m.

At the same large distance, $E_\text{axis} = 2E_\text{eq}$. A dipole field falls as $1/r^3$.

Dipole in a uniform field: net force zero; torque $\vec\tau = \vec p \times \vec E$, $\tau = pE\sin\theta$. Stable equilibrium at $\theta = 0^\circ$, unstable at $180^\circ$, maximum torque $pE$ at $90^\circ$. In a non-uniform field there is also a net force (towards stronger field when $\vec p$ is along $\vec E$).

Charge spread over an object is described by a density:

• linear $\lambda = dq/dl$ (C m$^{-1}$), e.g. a charged wire or ring;
• surface $\sigma = dq/dS$ (C m$^{-2}$), e.g. a charged plate or shell;
• volume $\rho = dq/dV$ (C m$^{-3}$), e.g. a charged cloud.

Set-up: split into elements $dq$, use $d\vec E = k\dfrac{dq}{r^2}\hat r$ for each, and add vectors: $\vec E = k\int \dfrac{dq}{r^2}\hat r$. Use symmetry to see which components cancel.

Flux through a small area: $\Delta\phi = \vec E\cdot\Delta\vec S = E\,\Delta S\cos\theta$, where $\Delta\vec S$ is normal to the area (outward for a closed surface). Unit N m² C$^{-1}$; a scalar.

Gauss’s law: $\displaystyle\oint_S \vec E\cdot d\vec S = \dfrac{q_\text{enc}}{\varepsilon_0}$.

• True for any closed surface of any shape.
• $\vec E$ in the integral is the total field (from charges inside and outside), but only enclosed charge contributes net flux.
• Charge outside: net flux zero. Charge inside: flux independent of the surface's size and shape and of where the charge sits inside.
• A charge $q$ at the centre of a cube sends $q/6\varepsilon_0$ through each face.

Steps for a derivation: draw the Gaussian surface; state why $\vec E$ is constant and normal on part of it and why other parts carry no flux; write flux $=$ charge enclosed $/\varepsilon_0$; solve for $E$; give the direction.