D.2 Electric and magnetic fields

Use the charge signs

There are two types of electric charge. Like charges repel: positive–positive and negative–negative. Unlike charges attract: positive–negative. The force on each charge acts along the line joining the two charges, with equal magnitude and opposite direction.

Draw the interaction

For two like point charges, draw arrows away from each other. For two unlike point charges, draw arrows toward each other. The direction is determined by the sign combination; the force magnitude also depends on charge magnitudes and separation, which Coulomb’s law quantifies.

Extend to a third charge

If a third charge is present, find the force from each other charge separately and add the force vectors. Do not decide the net direction by charge sign alone: compare the individual vectors and their magnitudes.

Common trap

Do not say that a negative charge always repels or that a positive charge always attracts. Attraction and repulsion depend on the pair of charges, and Newton’s third-law pair acts on different charges.

Use the inverse-square law

For two point charges, $r$ is their centre-to-centre separation. In a medium of permittivity $\varepsilon$, $k=1/(4\pi\varepsilon)$; in vacuum, $k=8.99\times10^9\,\mathrm{N\,m^2\,C^{-2}}$. Calculate magnitude, then use charge signs to state attraction or repulsion.

Worked example — unlike charges in air

For $q_1=4.5\times10^{-8}\,\mathrm{C}$, $q_2=-1.3\times10^{-7}\,\mathrm{C}$ and $r=3.2\times10^{-2}\,\mathrm{m}$, $F=(8.99\times10^9)|q_1q_2|/r^2=5.1\times10^{-2}\,\mathrm{N}$. The force is attractive because the charges have opposite signs.

Read the scaling

Doubling either charge doubles the force. Doubling the separation reduces the force to one quarter. If the medium has permittivity $\varepsilon$ rather than $\varepsilon_0$, use $k=1/(4\pi\varepsilon)$; greater permittivity reduces the force for the same charges and separation.

Choose the point-charge model

Spherical charged bodies can be treated as point charges at their centres when the geometry permits. Use centre-to-centre separation and convert charge units, such as microcoulombs, before substitution. For several charges, calculate each force vector and add them.

Common trap

Do not use diameter or a single radius as r, and do not forget that a change in separation is squared. Keep the force magnitude positive in the calculation, then state attraction or repulsion separately.

Core idea

Electric charge is conserved: in an isolated system, the total charge before an interaction equals the total charge after it. Charge can move between objects, but it is not created or destroyed in the transfer.

Use it at a junction

In a steady circuit, charge does not accumulate at a junction. The current entering equals the current leaving, for example $I_1=I_2+I_3$. This is a consequence of charge conservation, not a separate rule that overrides it.

Track the system boundary

When charge appears to change on one object, include the other object, the conductor or the ground in the system. Electrons may move across the chosen boundary, so the object’s charge changes while the total charge of the larger isolated system remains constant.

Common trap

Do not answer “Kirchhoff’s law” alone when asked for the fundamental law behind current balance. State conservation of electric charge.

Set the force balance

Millikan observed charged oil drops between parallel plates. By adjusting the potential difference, the electric force on a drop can balance its weight so the drop is stationary. With $E=V/d$, the balance is
$$qE=mg\quad\Rightarrow\quad q=\frac{mg}{E}=\frac{mgd}{V}$$
for the simplified model in which buoyancy is neglected.

Read the evidence

Repeating the measurement for many drops gives charges that are integer multiples of a smallest value, the elementary charge $e$: $q=ne$, where $n$ is an integer. This pattern is evidence that electric charge is quantized rather than continuously variable.

Explain the method

The experiment varies the electric field until a drop is held stationary, then uses the known mass and field to infer its charge. It is the repeated integer-multiple pattern—not one isolated drop—that supports the quantization conclusion.

Common trap

Do not say that Millikan directly measured a continuous range of charge or that the drop is uncharged when it is stationary. Stationary means the electric and gravitational forces balance; the charge is non-zero and can be calculated from the balance.

Transfer by friction

Rubbing two insulating materials can move electrons from one surface to the other. One object becomes negatively charged and the other positively charged; the total charge of the pair is conserved. The material that loses electrons is positive, and the material that gains electrons is negative.

Transfer by induction

Bring a charged object near a conductor without touching it. Charges in the conductor separate by repulsion and attraction. If the conductor is connected to ground while the charged object remains nearby, electrons can flow to or from Earth. Disconnect the ground first, then remove the external charged object, leaving the conductor with a net charge.

Transfer by contact and grounding

Touching a charged conductor to another conductor allows charge to redistribute between them. Grounding connects an object to a very large charge reservoir: electrons can leave an object or enter it, depending on the nearby charge and the object’s potential.

Common trap

Induction does not require contact with the charged rod. In a grounding sequence, remove the ground before removing the inducing charge; reversing the order can leave the conductor neutral.

Define the field

Electric field strength is force per unit positive test charge. For a point source $Q$, it is directed away from positive $Q$ and toward negative $Q$. Its SI unit is $\mathrm{N\,C^{-1}}$.

Worked example — point-charge field

At $r=1.0\,\mathrm{m}$ from $Q=+2.9\times10^{-8}\,\mathrm{C}$, $E=(8.99\times10^9)(2.9\times10^{-8})/(1.0)^2=2.6\times10^2\,\mathrm{N\,C^{-1}}$. Because the source is positive, the field points radially outward.

Add fields as vectors

For more than one source, calculate each electric-field vector at the point and add them. Do not add magnitudes unless all field vectors point in the same direction. The force on a particular charge is then $F=qE$, with its direction reversed from the field if the charge itself is negative.

Common trap

The field direction is defined using a positive test charge, not the sign of the test charge in the question. Also distinguish field strength $E$ from force $F$: changing the test charge changes F but not the source field E.

Interpret a field line

Electric field lines show the direction of the force on a small positive test charge. Their arrows point away from positive charges and toward negative charges. The tangent to a line gives the local field direction.

Read qualitative strength

Where field lines are closer together, the field is stronger; where they spread out, it is weaker. This is a qualitative representation unless the diagram specifies equal field-line intervals or a scale.

Common trap

Do not point electric field lines from negative to positive, make them cross, or draw them tangent to equipotential surfaces. Field lines follow the positive-test-charge convention.

Core idea

In a field-line diagram, greater line density represents a stronger electric field. Compare density over equal areas or equal widths of the same diagram; the visual spacing is a qualitative encoding of $|E|$, not a new physical force.

Connect density to distance

Around an isolated point charge, field lines spread as distance increases, so the field becomes weaker. A denser pattern near the charge is consistent with the inverse-square dependence of field strength. In a uniform field, equal spacing indicates constant field strength.

Check the representation

Density comparisons are meaningful only when the diagram uses the same line convention and potential/field intervals. Do not infer exact numerical values from arbitrary artwork; use labels or a scale if a calculation is required.

Common trap

Do not count field lines as individual objects or compare the total number of lines in two drawings with different scales. It is the local density that represents relative field strength.

Use the uniform-field model

Between two large opposite parallel plates, away from the edges, field strength equals potential difference $V$ divided by perpendicular plate separation $d$. The field points from the positive plate to the negative plate; edge regions are not uniform.

Worked example — required potential difference

For $E=1.0\times10^6\,\mathrm{V\,m^{-1}}$ and $d=0.50\,\mathrm{cm}=5.0\times10^{-3}\,\mathrm{m}$, $V=Ed=(1.0\times10^6)(5.0\times10^{-3})=5.0\times10^3\,\mathrm{V}$. The result applies to the uniform central region.

Read the direction

Electric field lines point from the positive plate to the negative plate, so a positive charge accelerates in that direction and a negative charge accelerates oppositely. The field strength can be expressed in $\mathrm{N\,C^{-1}}$ or equivalently $\mathrm{V\,m^{-1}}$.

Check the boundary

The formula assumes a uniform region and neglects edge effects. Use the perpendicular plate separation in metres; do not use the diagonal distance or the plate length.

Common trap

Do not reverse the field direction because the test charge is negative. Field direction is defined by a positive test charge; the force on a negative charge is opposite.

Interpret a magnetic field line

Magnetic field lines show the local direction of the magnetic field; a compass north pole or a suitable test direction follows the arrow convention. Unlike isolated electric field lines, magnetic field lines form continuous closed loops.

Use the right-hand rule

Around a long straight current-carrying wire, the field lines are concentric circles centred on the wire. Point the right thumb in the conventional current direction; curled fingers give the magnetic-field direction. If electrons move into the page, conventional current is out of the page, so reverse the electron-motion direction before applying the rule.

Common trap

Do not use electron motion as though it were conventional current, and do not draw magnetic field lines starting or ending on an isolated magnetic pole.

Set the reference

Electric potential energy of a system is the work done to assemble its charges from infinite separation to their present positions. Define the energy as zero at infinite separation. The sign depends on the charge combination: bringing opposite charges together lowers the energy, while bringing like charges together raises it.

Interpret energy transfer

Work done by an external agent changes the electric potential energy. If a positive charge moves toward a negative source, the electric field can do positive work while the potential energy decreases. If an electron is accelerated through a potential difference, the lost electric potential energy can become kinetic energy.

Track the system

For several charges, electric potential energy belongs to the whole charge configuration and is built from pair interactions. State the reference and the system before assigning a sign; do not confuse the energy of a charge configuration with electric potential at one point.

Common trap

Electric potential energy is not always positive. Unlike charges have negative pair energy relative to infinity; like charges have positive pair energy. The sign is determined by the interaction and reference, not by whether a charge is an electron.

Use the pair-energy expression

For two point charges or spherical conductors represented at their centres, keep the signs of $q_1$ and $q_2$ and use centre separation $r$. The zero reference is infinite separation.

Worked example — two negative conductors

Radii $2.5\,\mathrm{cm}$ and $1.5\,\mathrm{cm}$, separated by a $1.7\,\mathrm{cm}$ surface gap, give $r=5.7\times10^{-2}\,\mathrm{m}$. For charges $-4.7\times10^{-8}\,\mathrm{C}$ and $-6.3\times10^{-8}\,\mathrm{C}$, $E_p=(8.99\times10^9)q_1q_2/r=+4.7\times10^{-4}\,\mathrm{J}$. Positive energy matches repulsion between like charges.

Interpret the sign

Like charges give positive potential energy because work is required to bring them together. Unlike charges give negative potential energy because the electric field releases energy as they approach. Increasing separation moves the energy toward zero.

Use changes in energy

When a charge configuration changes, calculate the final minus initial potential energy. The external work and work done by the electric field have opposite signs under a quasistatic convention. Use the actual separation between charge centres, not the physical radius of either object.

Common trap

Do not use absolute values for q1q2 before deciding the sign of the energy, and do not put r² in the potential-energy formula; r² belongs to Coulomb force.

Define electric potential

Electric potential $V_e$ at a point is the work done per unit positive test charge in bringing it from infinity to that point. Its unit is $\mathrm{J\,C^{-1}}$, equivalent to volts. Set $V_e=0$ at infinity.

Add potentials algebraically

Electric potential is a scalar, so contributions from several point charges add with their signs:
$$V_e=\sum_i k\frac{Q_i}{r_i}$$
There is no vector-angle calculation when combining potential values.

Interpret the result

A positive source contributes positive potential and a negative source contributes negative potential. A point can have zero net potential because positive and negative contributions cancel, even though the electric field there is not necessarily zero.

Common trap

Do not add electric field magnitudes as though they were scalar, and do not infer zero electric field from zero potential. Potential and field are different quantities.

Use point-charge potential

Electric potential is signed and has zero at infinity. For source charge $Q$, use distance $r$ from the charge centre. Positive $Q$ gives positive potential; negative $Q$ gives negative potential.

Worked example — negative source

For $Q=-1.00\times10^{-8}\,\mathrm{C}$ at $r=1.00\,\mathrm{m}$, $V_e=(8.99\times10^9)(-1.00\times10^{-8})/(1.00)=-89.9\,\mathrm{V}$. At $2.00\,\mathrm{m}$, it is $-45.0\,\mathrm{V}$: farther away, the negative potential increases toward zero.

Combine sources

For several point charges, calculate each $kQ_i/r_i$ and add the scalar values. Use centre-to-point distance and convert all distances and charges before substitution. The potential does not depend on the test charge used to define it.

Check conducting spheres

Inside a charged conducting sphere in electrostatic equilibrium, the electric field is zero and the potential is constant throughout the interior. The potential need not be zero; it equals the surface potential for the ideal spherical case.

Common trap

Do not use $kQ/r^2$ for potential, and do not assume zero field means zero potential inside a conductor.

Use the gradient relationship

Electric field strength is the negative spatial gradient of electric potential. On a potential–distance graph, use the tangent gradient at the required point and state whether the answer is a signed component or a magnitude.

Worked example — tangent gradient

If a tangent changes from $-26\,\mathrm{kV}$ to $0\,\mathrm{V}$ over $8.0\,\mathrm{cm}=8.0\times10^{-2}\,\mathrm{m}$, $E=-[0-(-26\times10^3)]/(8.0\times10^{-2})=-3.3\times10^5\,\mathrm{V\,m^{-1}}$. The negative sign gives the field direction in the chosen coordinate.

Interpret the sign

The negative sign means the electric field points toward decreasing potential. A negative slope of $V_e$ against position corresponds to a positive field component in that coordinate direction; state whether the question wants a signed component or a magnitude.

Connect field to motion

A negative charge experiences force opposite to the electric field. Therefore its acceleration direction is opposite to the direction of decreasing potential, even though the field itself is always defined using a positive test charge.

Common trap

Do not use the graph’s potential value instead of its local gradient, and do not reverse the particle’s force direction without checking the particle’s charge sign.

Use potential difference

External work on a charge equals its change in electric potential energy. Keep the sign of $q$ and calculate final potential minus initial potential. Work by the field has the opposite sign.

Worked example — moving between equipotentials

A $+2.0\,\mathrm{C}$ charge moves from $40\,\mathrm{V}$ to $20\,\mathrm{V}$. Then $W_{\mathrm{on}}=(2.0)(20-40)=-40\,\mathrm{J}$. Its electric potential energy falls by $40\,\mathrm{J}$; if free, that energy can become kinetic energy.

Connect to kinetic energy

If only the electric field does work, $W_{\rm field}=-q\Delta V_e$, and the change in kinetic energy equals this work. A positive charge moving to lower potential can gain kinetic energy; a negative charge may gain kinetic energy moving to higher potential.

Use endpoints

Because electrostatic fields are conservative, the work between two points does not depend on the path. Motion along an equipotential has $\Delta V_e=0$ and therefore zero work by the field.

Common trap

Check which agent’s work the question asks for and keep the charge sign. Do not assume that moving a negative charge toward lower potential necessarily increases its kinetic energy.

Define an equipotential

An electric equipotential surface joins points with the same electric potential. Moving a charge along one surface gives $\Delta V_e=0$, so the electric field does no work.

Read spacing carefully

If equal potential intervals are drawn, closer equipotential lines indicate a larger potential gradient and stronger electric field. Use labels and the drawing convention; arbitrary visual spacing alone does not provide a numerical field.

Common trap

Do not confuse equipotential lines with field lines. A charge can move along an equipotential without field work, even though the electric field may be non-zero perpendicular to the path.

Use the perpendicular relationship

Electric field lines cross equipotential surfaces at 90°. The field points toward decreasing electric potential, so the field-line arrow is normal to the equipotential and in the direction of the negative potential gradient.

Apply it between plates

For oppositely charged parallel plates, field lines are approximately straight and perpendicular to the plates; equipotential surfaces are parallel to the plates. This is why the potential changes across the separation but remains constant along a plate.

Predict motion

A positive charge accelerates along the electric field, toward lower potential. A negative charge accelerates opposite to the field, toward higher potential. The field direction and particle-force direction must be kept separate.

Common trap

Do not draw field lines parallel to equipotentials or assume a negative charge accelerates in the field direction. The field is defined using a positive test charge.