Electric potential and capacitors: worked exercises

End of September 2026: in Year 5 of the liceo linguistico (Italian high school specialising in modern languages), electrostatics has just started, with charge and Coulomb’s law. In October the teacher moves on to electric field and potential, and the test on this block typically falls in November, between five and eight weeks from now. It is the same block that you meet in Year 5 at the classical high school and the human sciences high school with the same approach (charges, Coulomb, field, potential; Gauss almost always left out), and that at the scientific high school is tackled a year earlier, in Year 4, with one more hour a week and a few more calculations.

There are no derivatives or integrals here, and Gauss’s theorem almost certainly does not appear in the programme. What matters is keeping four quantities clearly apart: force, field, potential energy and potential. Their names are similar and their substance is different, two being vectors and two scalars. If you are starting electrostatics with Coulomb and force, this post carries on from there: field, potential energy, potential, and finally capacitors.

The electric field: force per unit charge

The electric field produced by a charge $Q$ at a point at distance $r$ is defined as the force that $Q$ would exert there on a test charge $q$, divided by $q$ itself:

For a point charge, substituting Coulomb’s law gives

The field is a vector: it exists at every point of space around $Q$, whether or not there is a second charge to “feel” it. It is represented by field lines, which leave positive charges and enter negative ones, and are closer together where the field is stronger.

A case that comes up often in problems is the uniform field, the one created between two parallel plates carrying opposite charges: here the field has the same magnitude, the same direction and the same sense at every point between the plates, from the positive plate towards the negative one.

Electric potential energy: stored work

Bringing a charge $q$ close to a charge $Q$ (of the same sign or the opposite sign) requires or releases energy, exactly like lifting a mass in the gravitational field. This energy is called electric potential energy:

Unlike the magnitude of the Coulomb force, here the signs of the charges must be kept, not replaced by absolute values. If $q$ and $Q$ have the same sign, $U$ is positive (the charges repel each other, and energy is needed to bring them closer). If they have opposite signs, $U$ is negative (they attract each other, and energy must be supplied to separate them).

Electric potential: energy per unit charge

The electric potential at a point is the potential energy that a test charge $q$ would have there, divided by $q$:

For a point charge $Q$:

Potential, unlike field and force, is a scalar quantity: it has a value at every point, but no direction. It is measured in volts ($\text{V}=\text{J/C}$).

The difference between potential and potential energy (the mistake everyone makes)

This is where almost every student trips up: the potential energy $U$ belongs to a specific charge (how much energy that charge $q$ has at that point), while the potential $V$ is a property of the point in space, which exists even before you put a charge there to “feel” it, just as the field $E$ exists before the test charge. The relation between the two is always $U=qV$.

The work done by the electric field

When a charge $q$ moves between two points at different potentials, the electric field does work:

where $\Delta V=V_A-V_B$ is the potential difference between the starting point $A$ and the arrival point $B$. The sign matters: if $q$ is positive and moves towards a lower potential ($\Delta V>0$), the work is positive, because the field “pushes” the charge in that direction, exactly as gravity pushes a mass downwards. If instead the charge moves towards a higher potential, the work done by the field is negative, and an external agent is needed to make it move.

The uniform field: the direct link between $E$ and $\Delta V$

In the particular case of the uniform field between two parallel plates at distance $d$, there is a simple relation between field and potential difference:

This is the formula that appears most often in problems with parallel-plate capacitors: knowing the voltage applied to the two plates and their distance, you immediately get the field between them, without going through the charge.

The electronvolt

For elementary charges (electrons, protons) joules are very awkward units: the numbers always come with huge negative exponents. So we use the electronvolt (eV): the energy gained by a charge equal to that of the electron crossing a potential difference of 1 volt.

It is just a more convenient unit of measurement, not a new concept: the conversion always and only uses this factor.

Equipotential surfaces

An equipotential surface is the locus of points that all have the same potential. Equipotential surfaces are always perpendicular to the field lines at every point, and this has a practical consequence: moving along an equipotential surface costs no work, because $\Delta V=0$. For a point charge the equipotential surfaces are spheres concentric with the charge; in the uniform field they are planes parallel to the plates.

Capacitors: capacitance, energy, connections

A capacitor is a system of two conductors (the “plates”) which, when subjected to a potential difference $\Delta V$, stores a charge $Q$ proportional to it. The ratio between charge and voltage is called capacitance:

and it is measured in farads ($\text{F}=\text{C/V}$), although in practice real values are always microfarads or picofarads: the farad on its own is an enormous unit.

For the parallel-plate capacitor, two parallel plates of area $S$ at distance $d$, immersed in a material (dielectric) with relative permittivity ${\epsilon }_r$, the capacitance depends only on the geometry and the material, never on the charge or on the applied voltage:

where ${\epsilon }_0=8.85\times 10^{-12}\text{F/m}$ is the permittivity of free space. Inserting a dielectric between the plates (instead of vacuum or air, where ${\epsilon }_r\approx 1$) always increases the capacitance, because the material becomes polarised and reduces the effective field between the plates for the same charge.

The stored energy

A charged capacitor stores energy in the electric field between the plates:

Equivalently, using $Q=C\Delta V$, it can also be written as $U_C=\frac{1}{2}Q\Delta V$ or $U_C=\frac{Q^2}{2C}$: they are the same formula written in three ways, and it is useful to choose the one with the quantities you already know in the problem.

Capacitors in series and in parallel

With several capacitors connected, the equivalent capacitance combines with rules opposite to those for resistors:

• in parallel (the same two plates in common, the same $\Delta V$ across each): the capacitances add,

• in series (plates in cascade, the same charge $Q$ on each): the reciprocals add,

For just two capacitors in series the direct form $C_{eq}=\frac{C_1{C}_2}{C_1+C_2}$ is more convenient.

Summary: vector or scalar, which unit

Typical mistake: confusing potential and potential energy

This is the mistake I see most often: the potential $V$ does not depend on the charge you put there, it is a property of the point; the potential energy $U$ does. If the text says “the potential at point A is 100 V”, that sentence does not change whether you put a charge of 1 C or of 1 μC there: what changes is the potential energy of that charge, $U=qV$. The second frequent mistake concerns the signs in the work $L=q\Delta V$: if you are not sure, reason physically (a positive charge “falls” towards lower potentials, as a mass falls downwards) instead of applying the formula by rote.

Three worked exercises

1. An electron accelerated by a potential difference. An electron (charge $e=1.6\times 10^{-19}\text{C}$, mass $m_e=9.11\times 10^{-31}\text{kg}$), initially at rest, is accelerated by a potential difference $\Delta V=500\text{V}$. Find the final speed.

The work done by the electric field on the electron is entirely converted into kinetic energy (the work-energy theorem):

I calculate the work:

I solve for $v$:

An enormous speed (about 4% of the speed of light), but consistent: electrons are extremely light and a few hundred volts are enough to accelerate them to a non-negligible fraction of the speed of light. At this value the classical treatment remains an excellent approximation, and for a Year 5 problem it is the one you use.

2. Parallel-plate capacitor with a dielectric. A parallel-plate capacitor has square plates with a side of $10\text{cm}$, at a distance $d=2\text{mm}$, with a dielectric of ${\epsilon }_r=4$ between the plates. It is connected to a voltage $\Delta V=12\text{V}$. Find the capacitance, the charge, the electric field and the stored energy.

Area of the plates: $S=(0.1{)}^2=0.01{\text{m}}^2$.

Capacitance:

Stored charge:

Electric field between the plates (I use $E=\Delta V/d$ directly, without going through $Q$):

Stored energy:

3. Two capacitors, first in series then in parallel. $C_1=4\mu \text{F}$ and $C_2=12\mu \text{F}$. Find the equivalent capacitance for the two connections.

In parallel:

In series:

Note that the equivalent capacitance in series ($3\mu \text{F}$) is smaller than the smaller of the two starting capacitors: it is the opposite of what happens with resistors, and in a test this plausibility check alone is enough to tell you whether you have swapped the two formulas.

Checked against a practice test pitched at Year 4

On the site you will find the electrostatics practice test, catalogued under Year 4 because at the scientific high school this block is done a year earlier. It is still good training on field, potential and capacitor problems. Some questions, in particular those on direct-current circuits, go beyond what you will see at the linguistico in Year 5, so do not be surprised if an exercise seems off-programme: concentrate on the pure electrostatics questions. As always: the correct answer is free, and the guided step-by-step solution is unlocked with the first lesson.

If you also want to revise the rest of the physics programme, you will find everything on the physics subject page. If you started from Coulomb and electric charge, the post before this one is Electrostatics and Coulomb’s law in Year 5, written for the art high school but with the same basic law. Anyone studying at the scientific high school and meeting this same block a year earlier, with circuits added, will find the right approach in Electrostatics and circuits: preparing for the Year 4 test, and for a review of all electromagnetism ahead of the maturità (Italy’s school-leaving exam) there is Electromagnetism for the maturità: the concept map for the final year. For the maths of the same year and track, the twin post is Maths in Year 5: functions and limits.

If you want a hand putting these four quantities in order before the November test, the first introductory call is free and lasts half an hour.