Electric Fields

An electric field is a region in which a charged object experiences a force. We measure how strong the field is at a point using the electric field strength, $E$, defined as the force per unit positive charge:

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

Here $F$ is the force on a small positive test charge $Q$ placed at that point. The unit is the newton per coulomb, $\text{N C}^{-1}$ (later you will see this is identical to the volt per metre, $\text{V m}^{-1}$).

Field strength is a vector. Its direction is defined as the direction of the force on a positive charge. So field lines point away from a positive charge and towards a negative charge. A negative charge placed in a field feels a force in the opposite direction to the field.

To find the force on any charge once you know the field, simply rearrange: $F = EQ$. A positive charge is pushed along the field; a negative charge is pushed against it.

Two point charges $Q_1$ and $Q_2$ separated by a distance $r$ exert a force on each other given by Coulomb's law:

$F = \frac{kQ_1Q_2}{r^2} = \frac{Q_1Q_2}{4\pi\varepsilon_0 r^2}$

where the Coulomb constant is $k = \frac{1}{4\pi\varepsilon_0} = 8.99\times10^{9}\ \text{N m}^2\text{ C}^{-2}, \qquad \varepsilon_0 = 8.85\times10^{-12}\ \text{F m}^{-1}$

Key features of this law:

• The force is inverse-square: double the separation and the force falls to a quarter.
• Like charges (both $+$ or both $-$) repel; opposite charges attract. A convenient rule: if $Q_1Q_2$ comes out positive the force is repulsive; if negative, attractive. In an exam it is usually clearer to work with the magnitudes of the charges and state the direction (attract/repel) in words.
• The forces on the two charges are equal and opposite (Newton's third law), even if the charges have very different sizes.

Notice the structural similarity to Newton's law of gravitation, $F = \dfrac{Gm_1m_2}{r^2}$: both are inverse-square, with $k$ in place of $G$ and charge in place of mass. The crucial difference is that gravity is always attractive, whereas the electric force can be either attractive or repulsive.

A point charge $Q$ produces a radial field — the field lines spread out in straight lines like the spokes of a wheel. Combining the definition $E = F/Q$ with Coulomb's law gives the field strength a distance $r$ from a point charge:

$E = \frac{kQ}{r^2}$

This is a vector and falls off as the inverse square of distance. For a positive $Q$ the field points radially outward; for a negative $Q$ it points radially inward.

The electric potential at that point — the work done per unit charge in bringing a small positive charge from infinity to that point — is

$V = \frac{kQ}{r}$

This is a scalar (unit volt, $\text{V} = \text{J C}^{-1}$) and falls off as $1/r$, more slowly than the field. Because potential is a scalar you add the potentials from several charges algebraically (with signs), which is far easier than adding field vectors.

Equipotentials are surfaces of constant potential. For a point charge they are concentric spheres (drawn as circles), and they are always perpendicular to the field lines. No work is done moving a charge along an equipotential.

Watch the difference in the powers of $r$: field strength $E\propto 1/r^2$ but potential $V\propto 1/r$. A handy check is that $E$ is (minus) the gradient of the $V$–$r$ graph, so the faster-falling quantity must be the field.

Two flat, parallel conducting plates a distance $d$ apart, connected to a potential difference $V$, produce a uniform field in the gap: the field lines are parallel, equally spaced, and point from the positive plate to the negative plate. The field strength is the same everywhere between the plates (edge effects aside) and is given by

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

This is where the alternative unit $\text{V m}^{-1}$ for field strength comes from. A charge $Q$ placed between the plates feels a constant force

$F = EQ = \frac{QV}{d}$

directed along the field (for a positive charge) — the same at every point in the gap, unlike the radial field which weakens with distance.

Radial vs uniform — the key contrast. A point charge gives a radial field where $E=kQ/r^2$ varies with position (inverse-square). Parallel plates give a uniform field where $E=V/d$ is constant. Choosing $kQ/r^2$ for a parallel-plate problem, or $V/d$ for a point charge, is a classic scope error — match the equation to the field geometry.

Fire a charged particle (say an electron) into the gap between charged plates, moving parallel to the plates. The particle experiences a constant force $F=EQ=\dfrac{QV}{d}$ perpendicular to its initial velocity. This is exactly the structure of projectile motion, with the uniform electric force playing the role of gravity:

• Along the plates (call it $x$): no force, so the horizontal velocity $u$ is constant → $x = u t$.
• Across the plates ($y$): a constant acceleration $a = \dfrac{F}{m} = \dfrac{EQ}{m} = \dfrac{QV}{md}$ → $y = \tfrac{1}{2}a t^2$.

The two motions share only the time $t$, and combining them gives a parabolic path — the particle curves towards the oppositely charged plate.

The method: find the time in the field from the horizontal motion ($t = x/u$), then feed that $t$ into the vertical equation $y=\tfrac12 a t^2$ to get the deflection. Because the deflection depends on $Q/m$, this arrangement can be used to compare or separate charged particles.

Electric fields (Unit 4) and gravitational fields (Unit 5) are deliberately taught with parallel mathematics — the same inverse-square structure — so it pays to line them up:

Feature	Electric field	Gravitational field
Force law	$F=\dfrac{kQ_1Q_2}{r^2}$	$F=\dfrac{Gm_1m_2}{r^2}$
Field strength	$E=\dfrac{F}{Q}=\dfrac{kQ}{r^2}$	$g=\dfrac{F}{m}=\dfrac{GM}{r^2}$
Potential	$V=\dfrac{kQ}{r}$	$V_{grav}=-\dfrac{GM}{r}$
Source of field	electric charge	mass
Constant	$k=\dfrac{1}{4\pi\varepsilon_0}$	$G$
Sign of force	attractive OR repulsive	always attractive

The similarities: both are inverse-square, both have a field strength defined as force per unit "test" quantity (per unit charge / per unit mass), and both share the pattern of $1/r^2$ for the field and $1/r$ for the potential.

The key differences:

• Charge comes in two signs, so the electric force can attract or repel; mass is always positive, so gravity only ever attracts.
• The electric force is enormously stronger — between two protons it is around $10^{36}$ times the gravitational force — which is why gravity is negligible in atomic physics but dominant on astronomical scales.

Capacitance — a quick forward look. Charged parallel plates store charge. The capacitance $C$ measures how much charge they hold per volt applied: $C = \frac{Q}{V}$ measured in farads ($\text{F}$, i.e. $\text{C V}^{-1}$). You will meet capacitors in full — energy stored, charging and discharging — in the Capacitance subtopic; here just note that the field between the plates you have been analysing is what stores the energy.