Coulomb’s Law at Home: From Two Balloons to a Digital Balance
Coulomb’s law can look like one of those areas of A-level Physics that requires complicated equipment, carefully controlled laboratory conditions and forces far too small to measure outside a specialist laboratory.
The equation itself may appear intimidating:
F = (1 / 4πε₀) × (|q₁q₂| / r²)
It is more commonly written as:
F = k × |q₁q₂| / r²
where:
F is the electrostatic force, measured in newtons;
q₁ and q₂ are the two charges, measured in coulombs;
r is the centre-to-centre distance between the charges, measured in metres;
k is the Coulomb constant, approximately 8.99 × 10⁹ N m² C⁻².
The law tells us that the force between two charged objects is proportional to the product of their charges and inversely proportional to the square of the distance between them.
In simpler language:
larger charges produce a larger force;
increasing the distance reduces the force;
like charges repel;
unlike charges attract.
However, the main idea can be demonstrated with two balloons, two pieces of string and an old woollen jumper.
With a little more equipment, including an ordinary digital mass balance, the demonstration can even be developed into a useful quantitative A-level investigation.
What Does the Inverse-Square Relationship Mean?
The most important part of Coulomb’s law is often the term:
1 / r²
This tells us that electrostatic force falls very rapidly as the distance between the charges increases.
For example:
if the distance is doubled, the force becomes one-quarter as large;
if the distance is tripled, the force becomes one-ninth as large;
if the distance is increased by a factor of four, the force becomes one-sixteenth as large.
This can be summarised as:
F ∝ 1 / r²
The symbol ∝ means “is proportional to”.
This is the same mathematical pattern found in Newton’s law of gravitation. The important difference is that gravitational forces are always attractive, while electrostatic forces can be attractive or repulsive.
The inverse-square relationship is easy to state. The challenge is finding a way to make it visible.
Investigation One: Demonstrating Coulomb’s Law with Balloons
Materials
You will need:
two similar rubber balloons;
two lengths of nylon thread or light string;
a woollen jumper, fleece blanket or dry hair;
a ruler or tape measure;
a fixed support from which the balloons can hang;
a plain wall or large sheet of paper to provide a background;
optionally, a phone or camera to record the movement.
This is mainly a qualitative demonstration. It clearly shows electrostatic repulsion, but it does not independently prove the inverse-square law.
Step 1: Prepare the Balloons
Inflate both balloons to approximately the same size and tie them securely.
Attach an equal length of thread to each balloon. A length of between 50 centimetres and one metre usually works well.
Suspend both threads from the same support point, or from two points very close together. The balloons should be able to move freely without touching furniture, walls, clothing or people.
Before the balloons are charged, they should hang close together.
Step 2: Charge Both Balloons
Rub each balloon vigorously against the same woollen material or against dry hair for approximately 10 to 15 seconds.
Rubbing transfers electrons between the materials. Because both balloons are charged using the same materials and method, they should acquire charges of the same sign.
The balloons will normally become negatively charged because they have gained electrons.
Try not to touch the rubbed surfaces afterwards. Charge can escape through your body, particularly if your hands are damp.
Step 3: Observe the Repulsion
Allow the balloons to hang freely.
The balloons should move apart and settle at a new equilibrium position. Each balloon is being pushed away from the other because like charges repel.
There are now three important forces acting on each balloon:
its weight acting vertically downwards;
the tension in the string;
the electrostatic force acting approximately horizontally.
The balloons stop moving when these forces balance.
This is already an important demonstration. Students can see that electric charge produces a real force without the charged objects touching.
Step 4: Measure the Separation
Measure the distance between the approximate centres of the two balloons.
Centre-to-centre distance is important because Coulomb’s law uses the distance between the centres of the charge distributions, not simply the air gap between the nearest surfaces.
Place a ruler behind the balloons rather than trying to insert it between them. A photograph taken square-on against a measured background can provide a more reliable measurement without disturbing the apparatus.
Step 5: Observe the Separation Over Time
Record the separation immediately after charging. Repeat the measurement at regular intervals, perhaps every 10 or 20 seconds.
A suitable results table could include:
| Time after charging | Centre separation | Observation |
|---|---|---|
| 0 seconds | Measure immediately | Maximum repulsion |
| 10 seconds | Record distance | Balloons remain separated |
| 20 seconds | Record distance | Separation begins to fall |
| 40 seconds | Record distance | Balloons move closer |
| 60 seconds | Record distance | Much weaker repulsion |
As charge gradually leaks from the balloons, the repulsive force decreases and the balloons move closer together.
Humidity makes a considerable difference. Moist air and damp surfaces allow charge to escape more rapidly, so the experiment normally works best in a warm, dry room.
An Important Scientific Limitation
It would be tempting to say that the changing balloon separation proves:
F ∝ 1 / r²
Unfortunately, it does not.
As the balloons lose charge, both the force and the separation change. The values of q₁, q₂ and r are all changing at the same time.
The experiment successfully demonstrates that:
charged objects exert forces without touching;
like charges repel;
stronger charging usually produces greater separation;
the force becomes weaker as charge leaks away.
It does not provide a controlled quantitative test of the inverse-square relationship.
Recognising this limitation is an important part of A-level experimental physics. A successful investigation should not claim more than the evidence can support.
Turning the Balloon Demonstration into a Force Measurement
The balloon arrangement can be developed using mechanics.
Suppose two identical balloons hang symmetrically from a common support point.
Let:
m be the mass of one inflated balloon;
L be the distance from the suspension point to the centre of the balloon;
d be the separation between the centres of the balloons;
θ be the angle made by each string with the vertical.
The horizontal displacement of each balloon is approximately half the total separation.
Therefore:
sin θ = d / 2L
For one balloon in equilibrium, the vertical component of the string tension balances its weight:
T cos θ = mg
The horizontal component balances the electrostatic force:
T sin θ = Fₑ
Dividing the horizontal equation by the vertical equation gives:
Fₑ = mg tan θ
Therefore, if the mass, suspension length and separation are measured, an estimate of the electrostatic force can be calculated.
This makes an excellent connection between two areas of A-level Physics: force equilibrium from mechanics and electrostatic force from electric fields.
However, the result remains an estimate. A balloon is not a point charge, its charge will not be distributed perfectly uniformly and small air currents can affect its position.
Investigation Two: Measuring Electrostatic Force with a Digital Balance
The second method is much more quantitative.
A digital balance does not measure mass directly. It measures the downward force acting on its pan and converts that force into an apparent mass reading.
If an additional electrostatic force pushes down on an object mounted on the pan, the displayed mass increases.
If an electrostatic force pulls the object upwards, the displayed mass decreases.
The change in the balance reading can therefore be converted into force using:
F = Δm × g
where:
F is the electrostatic force in newtons;
Δm is the change in apparent mass in kilograms;
g is the gravitational field strength, approximately 9.81 N/kg.
Equipment Required
You will need:
a digital mass balance, preferably reading to at least 0.01 g;
two similar lightweight conducting spheres;
two insulating supports;
a rigid laboratory stand;
a ruler, metre rule or vernier calliper;
a controlled charging source;
an electrometer or charge sensor with a Faraday pail, where available;
appropriate connecting and earthing equipment;
a stable, draught-free working area.
Table-tennis balls covered smoothly with conductive aluminium tape can be used as lightweight spheres. Care should be taken to remove sharp edges and large wrinkles.
A device sometimes described informally as a “coulomb meter” is more commonly called a charge sensor or electrometer in a school laboratory.
Why Use Conducting Spheres?
Coulomb’s law is written for point charges. A point charge is an idealised object with negligible physical size.
Real experimental objects have dimensions.
Conducting spheres are useful because their geometry is well defined. When the spheres are sufficiently far apart, they can approximately behave as though their charges were concentrated at their centres.
The approximation becomes less reliable when the spheres are brought very close together.
Charge on a conductor can redistribute in response to the electric field produced by the other sphere. The measured force may then differ from the simple point-charge prediction.
Setting Up the Balance
Mount one conducting sphere above the balance pan using a short insulating rod.
The support should be:
sufficiently rigid to prevent movement;
light enough not to overload the balance;
electrically insulating;
securely attached so that it cannot fall.
Place the second sphere directly above the first on a separate insulated support.
The centres of the two spheres must be aligned vertically. Misalignment means that only part of the electrostatic force acts vertically and is measured by the balance.
Measure the centre-to-centre distance, not merely the surface gap.
For identical spheres:
centre-to-centre distance = surface gap + sphere diameter
Using symbols:
r = s + D
where:
r is the centre-to-centre distance;
s is the gap between the surfaces;
D is the diameter of one sphere.
Zeroing the Apparatus
Place the lower sphere and its support on the balance.
Switch on the balance and allow the reading to stabilise.
Tare the balance so that the display reads zero.
The balance should now show only the change caused by the electrostatic interaction, provided that nothing else moves or touches the apparatus.
Keep hands, clothing, phones, metal stands and other conductors in fixed positions. Nearby objects can become polarised and affect the electrostatic force.
Charging the Spheres
Charge both spheres with charges of the same sign.
If the upper sphere is positioned directly above the lower sphere, the repulsion pushes the lower sphere downwards. The apparent mass displayed by the balance should increase.
With opposite charges, attraction pulls the lower sphere upwards and the displayed value should decrease.
The most difficult control variable is the amount of charge.
Charge leaks away continuously, and repeating the same rubbing action does not guarantee that exactly the same quantity of charge is transferred each time.
For a stronger investigation, measure q₁ and q₂ for every reading rather than assuming that they remain constant.
Collecting the Results
Select a series of centre-to-centre distances.
For example:
0.08 m;
0.10 m;
0.12 m;
0.14 m;
0.16 m.
At each distance:
discharge the spheres completely;
set and measure the required separation;
recharge the spheres;
measure or record the charge on each sphere;
return the spheres to the apparatus;
record the balance change immediately;
repeat the reading;
calculate a mean value.
A suitable results table would be:
| r in metres | q₁ in nC | q₂ in nC | Δm in grams | F in newtons | 1 / r² |
|---|---|---|---|---|---|
Converting the Balance Reading into Force
Suppose the balance reading increases by:
Δm = 0.120 g
This must first be converted into kilograms:
Δm = 0.120 ÷ 1000
Therefore:
Δm = 0.000120 kg
This can also be written as:
Δm = 1.20 × 10⁻⁴ kg
The electrostatic force is:
F = Δm × g
Substituting the values:
F = 1.20 × 10⁻⁴ × 9.81
Therefore:
F = 1.18 × 10⁻³ N
This is a force of approximately:
0.00118 N
It may appear to be a tiny force, but it is large enough to produce a measurable change on a sensitive digital balance.
A balance with a resolution of 0.01 g can, in principle, detect a force change of approximately:
9.8 × 10⁻⁵ N
In practice, vibration, air movement and charge leakage may reduce the sensitivity of the experiment.
Testing the Inverse-Square Law
Method One: Keep the Charge Approximately Constant
If q₁ and q₂ remain constant, Coulomb’s law can be written as:
F = kq₁q₂ × 1 / r²
Plot:
F against 1 / r²
A straight line passing close to the origin would support the inverse-square relationship.
The gradient of the graph would be:
gradient = kq₁q₂
This method is simple, but its validity depends on keeping the charges approximately constant.
Method Two: Correct for Changes in Charge
If the charge varies between readings, calculate:
F / |q₁q₂|
Coulomb’s law predicts:
F / |q₁q₂| = k / r²
A graph of:
F / |q₁q₂| against 1 / r²
should produce a straight line with a gradient close to k.
This is a stronger analysis because it does not assume that the charges remain perfectly constant.
Method Three: Use a Logarithmic Graph
For constant charge:
F = K / r²
Here, K represents the constant value kq₁q₂.
Taking logarithms gives:
log F = log K – 2 log r
A graph of:
log F against log r
should have a gradient close to:
gradient = –2
This provides another method of testing whether the relationship follows an inverse-square law.
Identifying the Variables
Independent Variable
The centre-to-centre separation, r, between the charged spheres.
Dependent Variable
The electrostatic force calculated from the change in apparent mass:
F = Δm × g
Control Variables
Important control variables include:
the charge on each sphere;
the diameter of the spheres;
the sphere material;
vertical alignment;
humidity;
the time between charging and recording;
the position of nearby conductors;
the position of the balance;
temperature;
air movement.
Sources of Uncertainty
Charge Leakage
Charge begins escaping as soon as the spheres are charged.
Readings must therefore be taken quickly and in a consistent sequence.
Humidity
Moist air and damp insulating supports increase charge leakage.
A dry room normally produces more reliable results.
Distance Measurement
A common mistake is measuring the gap between the surfaces instead of the distance between the centres.
Because the force depends on 1 / r², even a small uncertainty in distance can create a significant uncertainty in the predicted force.
The approximate fractional uncertainty relationship is:
ΔF / F ≈ 2 × Δr / r
This means that the percentage uncertainty in the force caused by the distance measurement is approximately twice the percentage uncertainty in the distance.
For example, if the percentage uncertainty in r is 3%, its contribution to the percentage uncertainty in F is approximately:
2 × 3% = 6%
Charge Redistribution
When conducting spheres are very close, charge may no longer be distributed as though each sphere were an isolated point charge.
Very small separations may therefore produce systematic deviations from the expected straight-line graph.
Balance Stability
Digital balances are sensitive to:
draughts;
vibrations;
movement of the supporting table;
warm hands near the apparatus;
electrostatic effects on the balance casing;
changes in the positions of wires and supports.
A draught shield and a solid bench can make a considerable difference.
Human Position
Even the person taking the reading can influence a sensitive electrostatic experiment.
Stand in approximately the same position for every reading and avoid moving your hands near the charged spheres while the value is being recorded.
Improving the Experiment
Several improvements can produce better results:
use a camera to record the balance display remotely;
place a scale behind the spheres;
use a micrometer adjustment to change the separation;
use identical spheres with accurately known diameters;
measure the charge during every trial;
recharge before each measurement;
repeat readings and calculate means;
randomise the order of the distances;
take background readings with both spheres discharged;
use a humidity meter and record the room conditions;
discard readings taken after a spark or accidental contact.
Randomising the order of the distances is particularly useful.
If every reading is taken from the smallest distance to the largest distance, charge leakage may create an additional trend. Later readings may have smaller forces simply because more charge has escaped.
Safety
The two-balloon version is a low-risk home activity, although care should be taken with latex allergies, broken balloon fragments and young children.
The quantitative version should be treated as a supervised school or college practical.
Do not improvise a mains-powered high-voltage supply. Use only purpose-designed, current-limited educational equipment and follow the manufacturer’s instructions.
Keep Van de Graaff generators and other electrostatic equipment away from sensitive electronic devices.
People with pacemakers or other implanted electronic medical equipment should not participate without appropriate specialist guidance.
Always discharge electrostatic equipment correctly before touching, adjusting or storing it.
Why I Like This Experiment
What I like about this investigation is the way it develops.
It begins with something almost anyone can do at home. Two balloons move apart, even though there is no visible connection between them.
That immediately creates a question:
What is pushing them apart?
The digital balance then turns that invisible interaction into a number.
A change of only a few hundredths of a gram becomes a force measured in newtons. That force can be compared with distance, charge and the equation in the textbook.
The balloons create curiosity.
The balance creates evidence.
The investigation also teaches an important lesson about experimental science. Producing a result is not enough. We must ask whether the variables were controlled, whether the apparatus measured what we think it measured and whether the mathematical model applies to the real objects being used.
Conclusion: Making an Invisible Force Visible
Coulomb’s law does not have to remain an equation copied into a set of notes:
F = k × |q₁q₂| / r²
Two charged balloons can show that like charges repel. Their changing separation demonstrates that electrostatic forces can move real objects and become weaker as charge escapes.
A digital balance can take the investigation much further.
By converting a change in apparent mass into force using:
F = Δm × g
and comparing the measured force with:
1 / r²
students can investigate one of the fundamental relationships in physics.
The experiment may begin with a balloon, a jumper and a piece of string. It can end with equilibrium calculations, uncertainty analysis, logarithmic graphs and an experimental estimate of the Coulomb constant.
That is what makes practical physics so valuable.
It takes an invisible force and gives us something we can observe, measure, question and understand.

