Charge, attraction and repulsion

Electric charge is a conserved property of matter that causes an object to experience electric forces in an electric field. Charge comes in two signs: positive and negative. Like charges repel; unlike charges attract. A neutral object has equal total positive charge and total negative charge, so its net charge is zero.

In everyday electrostatics, the moving charges are almost always electrons. A negatively charged object has extra electrons. A positively charged object is short of electrons. Use that wording in explanations: in solids, protons normally stay fixed inside nuclei, while electrons act as the mobile charge carriers.

A charged object can attract a neutral conductor or insulator by causing polarization, which is a separation of charge within an object while the object remains neutral overall. The nearer separated charge has the opposite sign, so its attraction is stronger than the repulsion from the farther separated charge.

Coulomb’s law

Coulomb’s law is an inverse-square law for the electrostatic force between two point charges. For charged bodies treated as point charges,

The sign of $q_1 q_2$ tells you what kind of force it is: a positive product gives repulsion, and a negative product gives attraction. The force acts along the straight line joining the point charges. If direction matters, don’t leave a Coulomb’s-law answer as just a number.

Coulomb’s constant can be written as

In a material medium we use

A larger permittivity gives a smaller force for the same charges and separation. Air behaves extremely close to a vacuum for most school calculations, but materials such as water, paper or rubber have different permittivities, so Coulomb-force calculations may give different results in different media.

How permittivity changes Coulomb force for the same charges and separation.

Coulomb’s original torsion-balance work gave an experimental route to the inverse-square relationship. In a modern school version, two small conducting-coated spheres on insulating supports can be used: vary the separation and use a balance reading or sideways displacement as a force indicator. The atmosphere must be dry, since charges leak away easily, and high-voltage supplies need sensible handling. If the force indicator is proportional to $F$, a plot against $1/r^2$ should be linear.

Conservation and transfer of charge

Conservation of electric charge is the principle that the total electric charge of an isolated system remains constant. Charge may move from one object to another, or around a circuit junction, but it is not created or destroyed in the process. At a junction, this is why the total current into the junction equals the total current out.

The syllabus includes three ways to transfer charge.

• Charging by friction is charge transfer caused by rubbing two materials together so that electrons move from one surface to the other. The two objects finish with opposite signs of charge.
• Charging by contact is charge transfer caused by touching a charged object to another object. The initially uncharged object finishes with the same sign of charge as the charged object, although the original object’s charge is reduced.
• Electrostatic induction is charging a conductor without contact by using a nearby charged object to separate charges, then using grounding to allow one sign of charge to leave or enter.

Grounding or earthing is connecting an object to Earth with a conductor so that charge can flow between the object and Earth until they are at the same potential. For example, bring a negatively charged rod near a neutral conducting sphere. Electrons in the sphere are repelled. If the sphere is grounded while the rod remains nearby, electrons can flow to Earth. Removing the ground connection before removing the rod leaves the sphere positively charged. If the inducing rod were positive, electrons would be drawn from Earth and the sphere would finish negative.

Millikan’s experiment and quantized charge

Quantization is a property of a physical quantity that can take only discrete values rather than a continuous range. Millikan’s oil-drop experiment provides evidence that electric charge is quantized.

In Millikan’s method, tiny oil drops are charged and watched between two horizontal plates. First, the motion of a falling drop with no electric field is used to determine its weight, allowing for drag and buoyancy. Then a uniform electric field is adjusted until the same drop is stationary. At rest, the electric force balances the effective weight of the drop. Repeating this for many droplets shows that measured charges are always integer multiples of one smallest charge: the elementary charge.

The elementary charge is the magnitude of the charge of a proton or electron. Its exact value is

A proton has charge $+e$ and an electron has charge $-e$.

This experiment is also a useful nature-of-science story. Millikan’s result was close to the modern value, but later examination of the data selection raised questions about rejected measurements and experimental bias. The key physics conclusion remains: no smaller free charge than $e$ appears in ordinary matter. Other quantities you meet later are also quantized, for example atomic energy levels and angular momentum in simple atomic models.

Electric field strength

Electric field strength is a vector quantity equal to the force per unit positive test charge placed at a point. It is defined by

The direction of $E$ is the direction of the force on a positive test charge. Put a negative charge in the same field and it experiences a force in the opposite direction. The “test charge” is a thought experiment: in reality, any charge you place in a field also has its own field and can disturb the arrangement you are trying to measure.

For a point source charge,

The field points away from a positive source charge and towards a negative source charge. With more than one source charge, electric field strengths add as vectors. Along a line, take care with signs; off a line, use components or a scale vector diagram.

This shared language is one of the great benefits of field theory. Gravitational, electric and magnetic fields are physically different, but we describe them using field strength, field lines, potential and energy. The terminology lets you transfer ideas from one field to another without pretending that the underlying causes are identical.

Electric field lines and field-line density

An electric field line is an imaginary curve whose tangent gives the direction of the electric field at each point. The arrow on the line shows the direction of the force on a positive test charge.

Use these conventions when sketching and interpreting electric field lines:

• field lines leave positive charge and enter negative charge
• field lines never cross, because the field cannot have two directions at one point
• closer field lines represent a stronger electric field
• field lines meet conducting surfaces at 90°
• for a radial field, the lines spread out with distance, showing that the field weakens.

For a single point charge the field is radial: outward for $+Q$, inward for $-Q$. For two like charges, field lines do not connect one charge to the other and there may be a zero-field point between them. For equal like charges, that point is halfway between them. For two unlike charges, field lines run from the positive charge to the negative charge.

Electric field patterns can be mapped experimentally using small particles suspended in oil between shaped electrodes. A high potential difference makes the particles align with the field. This technique is qualitative: you sketch the pattern, note where lines are denser, and compare it with the accepted field pattern. Treat high-voltage supplies with care.

Fields near conductors and conducting spheres

A conductor is a material that contains mobile charge carriers able to move through it. In electrostatic equilibrium, surplus charge on a conductor lies on its outer surface. If there were a component of electric field along the surface, charges would move; therefore the electric field at a conducting surface is perpendicular to the surface.

For a single spherical conducting body, solid or hollow, the field outside is the same as if the total charge were concentrated at the centre of the sphere. The field outside is radial. Inside the conducting material, the electric field is zero. Inside a hollow conducting sphere, the field in the hollow interior is also zero, provided no charge is placed inside the cavity. This shielding behaviour is the idea behind a Faraday cage.

Uniform field between parallel plates

A uniform electric field is an electric field with the same magnitude and direction throughout a region. Between two large, oppositely charged parallel plates, the field is approximately uniform away from the edges: straight, parallel, equally spaced lines from the positive plate to the negative plate.

Near the edges, the field is not uniform. Edge effects are the weakening and curving of the field near the ends of the plates, where field lines spread out into the surrounding space. In IB sketches, show the central field as straight and equally spaced, and show the edge field curving outward rather than stopping suddenly.

For parallel plates,

This gives an equivalent unit for electric field strength: $\mathrm{V\,m^{-1}}$. So $\mathrm{N\,C^{-1}}$ and $\mathrm{V\,m^{-1}}$ are the same physical unit in different clothing.

The force on a charge in an electric field is

$F = qE$.

Work done by an electric field or by an external force in moving a charge through a potential difference is often found from

The electronvolt is an energy unit equal to the energy transferred when a charge of magnitude one elementary charge moves through a potential difference of one volt. Numerically, $1\,\mathrm{eV} = 1.602176634 \times 10^{-19}\,\mathrm{J}$. Work done in electric fields may therefore be given in joules or electronvolts; joules are SI, while electronvolts are convenient for individual particles.

Magnetic field lines

A magnetic field is a region in which a magnetic pole, magnet or moving charge experiences a magnetic force. A magnetic field line is an imaginary curve whose tangent gives the direction in which a north-seeking pole would tend to move.

Magnetic field-line rules are similar to electric ones, with one crucial difference: magnetic field lines form closed loops. Outside a bar magnet they are drawn from the north-seeking pole to the south-seeking pole; inside the magnet the loop continues back from south to north. The lines never cross, and a greater density of lines represents a stronger magnetic field.

For a bar magnet, the field is densest near the poles. Unlike poles close together produce field lines linking across the gap, showing attraction. Like poles close together produce a region between them where fields oppose; for equal magnets there is a null point on the symmetry line.

Magnetic field patterns in this topic are restricted to a bar magnet, a current-carrying straight wire, a current-carrying circular coil and an air-core solenoid. Iron filings can show the shape of the field, but not its direction. A plotting compass gives direction because its north-seeking end aligns with the local magnetic field.

For a long straight current-carrying wire, the magnetic field lines are circles centred on the wire. Use the right-hand grip rule: point the thumb of your right hand in the direction of conventional current, and your curled fingers show the direction of the magnetic field lines. The field line spacing increases farther from the wire, so the magnetic field becomes weaker with distance.

For a circular coil, the field through the centre resembles the field of a short bar magnet. Looking at one face of the coil, anticlockwise conventional current makes that face north-seeking; clockwise current makes it south-seeking. Increasing the current or the number of turns strengthens the field.

An air-core solenoid is a long coil with no magnetic core. Its field pattern is like that of a bar magnet: nearly uniform and strong inside the solenoid, weaker and spreading outside. The field is strengthened by increasing current or turns per unit length. Adding an iron core would strengthen it further, but the restricted syllabus pattern here is the air-core solenoid.

The four fundamental interactions are gravitational, electromagnetic, weak nuclear and strong nuclear. Electric and magnetic effects are parts of the electromagnetic interaction, which is vastly stronger than gravity at particle scales and has infinite range. Moving charges produce magnetic fields, and moving charges in magnetic fields can be forced into curved paths; that is the doorway to particle accelerators and probes of matter in later topics.

Four fundamental interactions compared by strength and range.

Electric potential energy

Electric potential energy is the energy stored in a system of charges because of where the charges are relative to each other. More exactly, it is the work an external agent does to assemble the system from infinite separation, while leaving the charges with no change in final kinetic energy.

For two point charges,

The sign is doing real work here. Two like charges have positive $E_p$: you must do work to push them together from infinity. Two unlike charges have negative $E_p$: the field can do work as they come together, so an external agent would need to remove energy to assemble them slowly.

For more than two charges, add the potential energies of all distinct pairs. Don’t count the same pair twice. Scalar energy is friendlier than vector force in this situation, because potential energies add algebraically.

Electric potential

Electric potential at a point is the work done per unit positive test charge in bringing the test charge from infinity to that point. It is scalar, so potentials from several charges add algebraically rather than as vectors.

The zero of electric potential is defined at infinity. This matches gravitational potential work from D.1 and keeps the language consistent for inverse-square fields. In circuits, Earth is often called “zero volts”, but that is just a local reference choice, not the absolute zero at infinity.

For a point charge,

A positive source charge gives positive potential; a negative source charge gives negative potential. Near a point charge, the magnitude of the potential is larger because $V_e$ depends on $1/r$.

The work done in moving a charge through a potential difference is

The same energy transfer may be written in joules or electronvolts. For example, a proton moving through a potential difference of 1 V changes its energy by 1 eV; an electron moving through the same potential difference changes by the same magnitude, but the sign depends on whether its electric potential energy increases or decreases.

Electric field strength as potential gradient

Electric field strength tells you how rapidly electric potential changes with position. In one dimension,

The minus sign is not just decoration. It shows that the electric field points in the direction of decreasing electric potential for a positive test charge. Release a positive charge and it accelerates “downhill” in potential; an electron accelerates opposite the field, although the same potential-gradient rule still describes the field.

For a uniform field between parallel plates, the potential-distance graph is a straight line, so the field is constant. For a point charge, the potential-distance graph follows $1/r$, while the electric field strength follows $1/r^2$. The gradient of the potential graph gives $-E$; the area under an electric-field-strength against distance graph gives the potential change, with the sign handled by the minus convention.

This relationship gives one of the clearest links between electric and gravitational fields. Force and potential energy depend on the charge or mass of the test object; field strength and potential remove that test-object factor, leaving a property of the field itself.

Equipotential surfaces

An equipotential surface is a surface on which every point has the same electric potential. Since $\Delta V_e = 0$ along an equipotential, no work is done in moving a charge along it. The same statement applies to a mass moving along a gravitational equipotential.

Electric field lines always meet equipotential surfaces at 90°. If an electric field had a component along an equipotential, a charge could move along it and gain or lose energy, contradicting equal potential. This gives a useful sketching rule: field lines cut equipotentials at right angles, and closer equipotentials mean a larger potential gradient and therefore a stronger field.

For a point charge, equipotentials are concentric spheres centred on the charge. In a two-dimensional diagram, they show up as concentric circles. They are not equally spaced for equal voltage steps because $V_e$ depends on $1/r$.

For a collection of point charges, up to four in IB recognition questions, equipotentials are found by adding the scalar potentials from each charge. The patterns can be distorted loops or saddle shapes. They never cross: one point cannot have two different potentials. The electric field is strongest where neighbouring equipotential lines are closest together.

Between oppositely charged parallel plates, ignoring edge effects, equipotential surfaces are planes parallel to the plates. In a two-dimensional side view, they appear as equally spaced lines parallel to the plates. The electric field lines are straight, equally spaced, and perpendicular to the equipotentials.

Equipotentials can be mapped experimentally with conducting paper. Copper foil electrodes are connected to a low-voltage supply, and a high-resistance voltmeter probe is used to find points at the same potential. Join the points to draw an equipotential line. Different electrode shapes model different charge arrangements; the accepted field pattern is then inferred by drawing field lines at right angles to the equipotentials.

Equipotentials for conducting spheres

For a charged conducting sphere, the surface is an equipotential. If different parts of the conductor were at different potentials, mobile charges would move until the potential difference vanished.

Outside a solid or hollow charged conducting sphere, the potential is the same as for a point charge at the centre with the same total charge. The equipotentials outside are concentric spheres. Inside the conducting material, $E = 0$, so the potential gradient is zero and the potential is constant.

For a hollow charged conducting sphere with no charge inside the cavity, the whole interior is also at the same potential as the surface. The electric field is zero throughout the hollow region, so no work is done moving a charge around inside. For a solid charged conducting sphere, the same constant-potential idea applies throughout the conducting interior.

This is electrostatic shielding described through potential. The conductor is not simply “blocking lines”; its mobile charges have rearranged until the internal electric field is zero and the whole conductor is an equipotential volume.

Mapping fields algebraically and visually

A field can be read in two complementary ways. Algebra gives exact relationships such as Coulomb’s law, $V_e = kQ/r$, and $E = -\Delta V_e/\Delta r$. Diagrams show direction, symmetry and relative strength quickly: field lines show force direction, while equipotentials show energy-per-charge structure.

Use both. Around a positive point charge, for example, the algebra says $V_e$ is positive and decreases with distance, while $E$ points radially outward and weakens as $1/r^2$. The diagram tells the same story: spherical equipotentials spread farther apart as you move away, field lines cross them normally, and field-line density decreases with distance.

The old planetary model of the atom borrowed this field language: electrons were imagined orbiting a positive nucleus like planets orbiting the Sun. It captures a central attractive interaction but fails as a full model because atoms have quantized energy states and electrons are not tiny planets following classical orbits. That failure matters historically: it shows why quantization, first hinted at here by charge and later by spectra, is not optional in atomic physics.