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.