Electric and Magnetic Fields

Electric charge is a fundamental property of matter that comes in two types, positive and negative. Charge is quantized, meaning every observed charge is a whole-number multiple of the elementary charge $e = 1.60 \times 10^{-19}\ \text{C}$, and it is conserved: charge can be transferred (for example by friction, when electrons move from one surface to another) but never created or destroyed.

Coulomb's law gives the force between two point charges:

$$F = \frac{q_1 q_2}{4 \pi \varepsilon_0 r^2}$$

The force is proportional to the product of the charges, falls off with the inverse square of the separation, and acts along the line joining the charges. The constant $k = \frac{1}{4\pi\varepsilon_0} \approx 8.99 \times 10^9\ \text{N m}^2\ \text{C}^{-2}$, where $\varepsilon_0$ is the permittivity of free space.

• Like charges repel; opposite charges attract.
• Doubling the separation cuts the force to one quarter.

For a sense of scale: two $1\ \mu\text{C}$ charges held $10\ \text{cm}$ apart push on each other with about $0.9\ \text{N}$, roughly the weight of an apple. You must be able to apply Coulomb's law numerically and add the forces from several charges as vectors (superposition).

An electric field is the region around a charged object in which another charge experiences a force. The electric field strength at a point is the force per unit charge on a small positive test charge placed there, with units $\text{N C}^{-1}$ (equivalently $\text{V m}^{-1}$):

$$E = \frac{F}{q}$$

Because force is a vector, so is $E$: it points away from positive charges and toward negative charges. Combining $E = F/q$ with Coulomb's law shows that the field of a point charge is radial and weakens with the inverse square of the distance.

Field lines are a visual map of the field:

• They start on positive charges and end on negative charges.
• The arrow gives the direction of the force on a positive charge.
• Line density shows field strength: lines bunch together where the field is strong.
• Field lines never cross, because the field has a single direction at each point.

An isolated positive point charge, for example, has lines radiating outward symmetrically in all directions. You should be able to sketch the patterns for single charges, pairs of like and unlike charges, and a charged sphere, and calculate net field strengths by vector addition.

Two parallel conducting plates connected to a power supply create a uniform electric field in the gap: the field has the same magnitude and direction everywhere between the plates (ignoring fringing at the edges). The field points from the positive plate to the negative plate, and its strength is given by:

$$E = \frac{V}{d}$$

Here $V$ is the potential difference and $d$ is the plate separation. Field lines are drawn as equally spaced parallel lines running perpendicular to the plates.

A charge $q$ in the gap feels a constant force $F = qE$, so it undergoes constant acceleration, just like a projectile in a gravitational field. A charged particle entering the gap sideways therefore follows a parabolic path; this is how older oscilloscopes and inkjet printers steer charged particles.

A classic experimental setup balances the electric force against weight: a charged droplet hangs stationary between horizontal plates when $qE = mg$.

You must be able to:

• Calculate $E$ from the supply voltage and separation.
• Find the force and acceleration on a charge in the gap.
• Recognise that the field strength does not depend on where the charge sits between the plates.

Magnetic fields are produced by magnets and by moving charges: every electric current has a magnetic field looping around it. A magnetic field is a region in which a magnetic material, a current-carrying conductor, or a moving charge experiences a force.

Field lines visualize the field. Outside a bar magnet they emerge from the north pole and enter the south pole; inside the magnet they run from south back to north, so every line is a continuous closed loop. This is a key contrast with electric field lines: there are no magnetic monopoles, so magnetic field lines never start or stop on anything.

The arrow shows the direction a small compass needle (or a free north pole) would point, and the line density indicates the field strength.

You should be able to sketch the standard patterns:

• A single bar magnet.
• Two magnets attracting or repelling.
• The concentric circles around a straight current-carrying wire (direction given by the right-hand grip rule).
• The bar-magnet-like field of a solenoid.

Iron filings scattered around a magnet line up along the field and make the pattern visible, a simple demonstration worth remembering when describing field shapes.

Electric potential $V_e$ at a point is the work done per unit positive charge in bringing a small test charge from infinity to that point. It is a scalar measured in volts ($1\ \text{V} = 1\ \text{J C}^{-1}$), which makes combining contributions from several charges easy: simply add the values, with no components needed. For a point charge:

$$V_e = \frac{q}{4 \pi \varepsilon_0 r}$$

Note the $1/r$ dependence rather than $1/r^2$, and that the sign of $V_e$ follows the sign of the charge.

The electric potential energy of two point charges is:

$$E_p = \frac{q_1 q_2}{4 \pi \varepsilon_0 r}$$

Do not confuse the two quantities: potential is energy per unit charge, while $E_p$ is an energy measured in joules. When a charge $q$ moves between two points, the work done by the field is $W = q\Delta V_e$; this is exactly how accelerating a particle through a potential difference gives it kinetic energy, as in an electron gun.

Field and potential are linked by $E = -\frac{\Delta V_e}{\Delta r}$, so a rapidly changing potential means a strong field. You must be able to calculate $V_e$ and $E_p$ for point-charge systems and use $W = q\Delta V_e$ in energy problems.

An equipotential surface joins all points that share the same electric potential. Because the potential does not change along the surface, zero work is done when a charge is moved anywhere on it, regardless of the path taken.

Around a point charge the equipotentials are concentric spheres; between parallel plates they are flat planes parallel to the plates. They behave like contour lines on a map: closely spaced equipotentials mean the potential is changing quickly, which signals a strong field.

The relationship between fields and potentials makes this precise. The electric field strength vector is the negative spatial gradient of the electric potential:

$$E = -\frac{\Delta V_e}{\Delta r}$$

This implies that electric field lines must always point in the direction of the steepest decrease in electric potential. Consequently, equipotential surfaces must cross electric field lines at exactly $90^{\circ}$: if there were a non-zero component of the electric field parallel to an equipotential surface, a force would act along the surface, and work would be required to move a charge along it, violating the definition of an equipotential.

You should be able to sketch equipotential patterns for point charges, charge pairs and parallel plates, read potential differences from them, and use the gradient relation to estimate field strength from equipotential spacing.