IMAT Physics · Formula sheet

Electricity formulas

13 formulas, 2 reference tables and 19 common traps for IMAT Physics Electricity, grouped by subtopic.

Electric Charge, Conductors and Coulomb's Law

Electric charge: carried by particles, quantised and conserved

Charge as a count of elementary charges

Q = n e

$Q$ charge, in C
$n$ number of electrons gained or lost (a whole number)
$e$ elementary charge, $1.6 \times 10^{-19}\ \text{C}$

Coulomb's law: the force between two point charges

Coulomb's law

F = k\,\frac{q_1 q_2}{r^2}

$F$ force on each charge, in N
$q_1, q_2$ the two charges, in C
$r$ distance between them, in m
$k$ $9.0 \times 10^9\ \text{N m}^2/\text{C}^2$

Electric field: force per unit charge, and field lines

Electric field

E = \frac{F}{q} \qquad E = k\,\frac{Q}{r^2}

$E$ electric field strength, in N/C
$F$ force on a charge $q$ placed in the field, in N
$Q$ the charge that makes the field, in C
$r$ distance from $Q$ , in m

Conductors and insulators: where charge can move

Material	Free charge carriers	Charge placed on it	Examples
Metal (conductor)	Electrons	Spreads over the outer surface	Copper, aluminium, gold
Electrolyte (conductor)	Positive and negative ions	Moves with the ions	Salt water, blood plasma, acid solutions
Semiconductor	A few electrons, more when warmer	Moves slowly	Silicon, germanium
Insulator	None	Stays where it was placed	Glass, plastic, rubber, dry air, pure water (nearly)

Most metals conduct better when cold; semiconductors conduct better when warm.

Common traps

A neutral atom has zero charge, not a small one

An atom has as many electrons as protons, so the charges cancel exactly. Its net charge is 0 C. Options giving a neutral atom a tiny charge in nC or pC are wrong; only an ion carries a net charge.

Positive charge means electrons lost, not protons gained

Protons are locked in nuclei and do not move from one solid to another. A body becomes positive by losing electrons and negative by gaining them.

Insulators do not share charge on contact

Sharing charge equally on contact is a property of conductors. If two insulating bodies touch, the charge cannot flow, so each keeps the charge it had. Options that average the charges by size or radius apply only to metals.

The bigger charge does not feel a bigger force

The forces on the two charges are a Newton's third law pair: equal in size and opposite in direction, whatever the two charges are. Only the accelerations can differ, because the masses can differ.

Distance enters squared

Coulomb's law is an inverse square law. Tripling the distance divides the force by 9, not by 3. Options that divide by the distance factor alone are the usual distractor.

Electrons move against the field

The field direction is defined by the force on a positive charge. An electron feels a force of the same size in the opposite direction. Options that send an electron along the field lines are wrong.

Electric Potential, the Electronvolt and Capacitors

Learn this subtopic in the notes

Potential difference, the work W = qV and the electronvolt

Work on a charge, and the uniform field

W = qV \qquad E = \frac{V}{d}

$W$ work done on the charge (energy gained), in J
$q$ charge, in C
$V$ potential difference crossed, in V
$d$ separation of the plates, in m

Capacitors: C = Q/V, stored energy, series and parallel

Capacitance, energy and combinations

C = \frac{Q}{V} \qquad U = \tfrac{1}{2}CV^2 \qquad C_{\parallel} = C_1 + C_2 \qquad \frac{1}{C_{\text{series}}} = \frac{1}{C_1} + \frac{1}{C_2}

$C$ capacitance, in F
$Q$ charge on one plate, in C
$V$ p.d. across the capacitor, in V
$U$ energy stored, in J

Common traps

The electronvolt is energy, not voltage

Despite its name, the eV measures energy:

1\ \text{eV} = 1.6 \times 10^{-19}\ \text{J}

. A particle with charge

2e

crossing 500 V gains 1000 eV, not 500 eV. Writing an eV value in joules without converting gives an answer about

10^{19}

times too big.

Plate spacing goes in metres

E = V/d

the spacing must be in metres. Using millimetres directly makes the field 1000 times too small.

Capacitors combine the opposite way to resistors

Capacitors in parallel add directly; in series you add reciprocals. Resistors do the reverse. Swapping the rules is the most common wrong answer in capacitor network questions.

Stored energy is half QV

The energy in a charged capacitor is

\tfrac{1}{2}QV

, not

QV

. The voltage builds up from zero while charging, so on average each bit of charge was pushed on at half the final voltage.

Current, Ohm's Law and Resistivity

Learn this subtopic in the notes

Electric current: charge per second, and which way it flows

Current

I = \frac{Q}{t}

$I$ current, in A
$Q$ charge passing a point, in C
$t$ time, in s

Resistance and Ohm's law

Ohm's law

V = I R

$V$ p.d. across the component, in V
$I$ current through it, in A
$R$ resistance, in Ω

Resistivity: how a wire's length and thickness set its resistance

Resistance of a wire

R = \frac{\rho L}{A} \qquad A = \frac{\pi d^2}{4}

$\rho$ resistivity of the material, in Ω m
$L$ length, in m
$A$ cross-section area, in m²
$d$ diameter of the wire, in m

Common traps

Electrons move against the conventional current

The arrow on a circuit diagram shows conventional current, from + to −. In a metal wire the charges actually moving are electrons, going the opposite way. Protons do not move along a wire at all, so any option with moving protons in a metal is wrong.

Ohm's law needs a constant resistance

R = V/I

defines resistance for any component at any moment, but Ohm's law (current proportional to voltage) holds only when the resistance is constant, as for a metal at steady temperature. A filament lamp does not obey it, so doubling its voltage less than doubles its current.

Diameter is squared in the area

The area of a round wire is

\pi d^2/4

. Doubling the diameter quarters the resistance, it does not halve it. Forgetting the 4 in

\pi d^2/4

, or using the diameter as the radius, gives an answer four times off.

Resistor Networks, Kirchhoff's Rules and Internal Resistance

Learn this subtopic in the notes

Resistors in series and in parallel

Series and parallel

R_{\text{series}} = R_1 + R_2 + \dots \qquad \frac{1}{R_{\parallel}} = \frac{1}{R_1} + \frac{1}{R_2} + \dots

$R_1, R_2$ the individual resistances, in Ω
$R_{\parallel}$ the single resistance equal to the parallel group

Kirchhoff's rules, the potential divider and meters

Potential divider

V_1 = V \, \frac{R_1}{R_1 + R_2}

$V$ total p.d. across the series chain, in V
$V_1$ p.d. across $R_1$

EMF and internal resistance of a battery

EMF and internal resistance

\varepsilon = I(R + r) \qquad V = \varepsilon - I r

$\varepsilon$ e.m.f. of the battery, in V
$R$ external (load) resistance, in Ω
$r$ internal resistance, in Ω
$V$ terminal p.d., in V

Common traps

A parallel total is smaller than every branch

Putting resistors in parallel gives the current extra paths, so the total is less than the smallest single resistor. If your parallel answer is bigger than one of the branches, you have added the resistances instead of their reciprocals.

Meters in the wrong place

An ammeter in parallel with a component short-circuits it, because the ammeter has almost no resistance. A voltmeter placed in series stops the current almost completely and reads close to the full supply voltage. Read each meter's position before using its reading.

Terminal p.d. is less than the e.m.f. when current flows

The e.m.f. is reached only when no current flows. Draw a current and the battery loses

Ir

inside itself, so the voltage across the circuit falls. The bigger the current (the smaller the load resistance), the bigger the drop.

Electrical Power, Energy and the Effects of a Current

Learn this subtopic in the notes

Electrical power: P = VI = I²R = V²/R

Electrical power

P = VI = I^2 R = \frac{V^2}{R}

$P$ power, in W
$V$ p.d. across the component, in V
$I$ current through it, in A
$R$ resistance, in Ω

Electrical energy and the kilowatt-hour

Electrical energy

E = P t = V I t \qquad 1\ \text{kWh} = 3.6 \times 10^6\ \text{J}

$E$ energy, in J (or kWh)
$P$ power, in W (or kW)
$t$ time, in s (or h)

The three effects of an electric current

Effect	Cause	Uses	In a superconducting wire?
Heating	Charges colliding with the conductor's ions	Kettles, toasters, fuses, filament lamps	Absent (zero resistance)
Magnetic	Any moving charge	Electromagnets, motors, relays, MRI scanners	Present
Chemical	Ions discharged at electrodes	Electrolysis, electroplating, charging batteries	Absent (no ions move in a solid metal)

Common traps

Doubling the current quadruples the power

In a fixed resistor

P = I^2 R

, so the power goes with the square of the current. Doubling the current multiplies the heating by 4, not 2. The options often include the result of forgetting the square.

The kilowatt-hour is energy, not power

A kWh is a kilowatt multiplied by an hour, so it measures energy, like the joule. Mixing watts with hours, or kilowatts with seconds, gives answers off by factors of 1000 or 3600.

Zero resistance does not remove the magnetic effect

A superconductor has no resistance, so it produces no heat. It still produces a magnetic field, because the field comes from the moving charge, not from the resistance. That is exactly why superconducting coils make the strongest electromagnets.

Electricity formulas

Electric Charge, Conductors and Coulomb's Law

Electric charge: carried by particles, quantised and conserved

Coulomb's law: the force between two point charges

Electric field: force per unit charge, and field lines

Conductors and insulators: where charge can move

Common traps

Electric Potential, the Electronvolt and Capacitors

Potential difference, the work W = qV and the electronvolt

Capacitors: C = Q/V, stored energy, series and parallel

Common traps

Current, Ohm's Law and Resistivity

Electric current: charge per second, and which way it flows

Resistance and Ohm's law

Resistivity: how a wire's length and thickness set its resistance

Common traps

Resistor Networks, Kirchhoff's Rules and Internal Resistance

Resistors in series and in parallel

Kirchhoff's rules, the potential divider and meters

EMF and internal resistance of a battery

Common traps

Electrical Power, Energy and the Effects of a Current

Electrical power: P = VI = I²R = V²/R

Electrical energy and the kilowatt-hour

The three effects of an electric current

Common traps

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