I = dQ/dt ; steady transfer: I = Delta Q / Delta t
Current equals the rate of change of charge with time; for steady transfer it is total charge divided by total time.
Net rate of charge crossing an oriented surface.
- I
- current (A = C/s)
- Q
- charge (C)
- t
- time (s)
Use when — Charge transfer through a cross-section is given or required.
Common trap — Confusing conventional current direction with electron drift direction.
I = integral(J dot dA) ; uniform and normal: I = JA
Current equals the surface integral of current density dotted with the area element; for uniform normal flow it is current density times area.
Local current per unit normal area, directed along conventional positive-charge flow.
- J
- current density (A/m^2)
- dA
- area element with chosen normal (m^2)
Use when — Cross-section, nonuniform flow or local material behaviour matters.
Common trap — Treating J as a scalar in a surface whose normal is not parallel to it.
J = n q v_d ; uniform wire magnitudes: I = n e A v_d
Current density equals carrier density times carrier charge times drift velocity; for a uniform wire, current equals n e A times drift speed.
Many carriers with a small average drift can produce a measurable current.
- n
- carrier number density (m^-3)
- q
- carrier charge (signed in the vector form) (C)
- v_d
- drift velocity (m/s)
- A
- cross-sectional area (m^2)
Use when — Connecting carrier density, area, drift speed and current.
Common trap — Dropping the sign of carrier charge in the vector form, or assuming every electron moves only with the drift speed.
mu = |v_d|/E ; mu = e tau/m ; J = sigma E ; rho = 1/sigma
Mobility is drift speed per unit field; in the simple electron model it equals e tau over m. Current density equals conductivity times field, and resistivity is the reciprocal of conductivity.
Mobility describes carrier response, conductivity describes material current response and resistivity describes opposition in a geometry-independent material model.
- mu
- mobility (m^2/(V s))
- sigma
- conductivity (S/m)
- rho
- resistivity (ohm m)
- tau
- mean relaxation time (s)
Use when — The material is in its linear-response range.
Common trap — Applying the one-carrier Drude formula unchanged to a system with several carrier types.
V = IR with constant R for an Ohmic conductor
Voltage equals current times resistance, with resistance constant for an Ohmic conductor under unchanged conditions.
Potential difference is proportional to current over the stated operating range.
- V
- potential difference (V)
- I
- current (A)
- R
- resistance (ohm)
Use when — Linearity and stable physical conditions are stated or established.
Common trap — Assuming that calculating V/I at one point proves Ohmic behaviour.
R = rho L / A ; varying area: dR = rho dx / A(x)
Resistance equals resistivity times length divided by area; for a varying area, integrate rho dx over A of x.
Specimen resistance combines material resistivity with length and cross-sectional area.
- R
- resistance (ohm)
- rho
- resistivity (ohm m)
- L
- length (m)
- A
- cross-sectional area (m^2)
Use when — Material and cross-section are uniform and physical conditions are specified.
Common trap — Using the uniform formula for a tapered wire without integration.
R_T = R_0 [1 + alpha (T - T_0)]
Resistance at temperature T equals resistance at the reference temperature times one plus alpha times the temperature difference.
Approximate linear change of resistance with temperature over a suitable range.
- alpha
- temperature coefficient (K^-1 or per degree C for intervals)
- T, T_0
- temperature and reference temperature, consistent intervals
Use when — A linear range and a reference value are appropriate.
Common trap — Treating alpha as universal, ignoring the reference temperature or applying a metal trend to a semiconductor.
R_eq = sum(R_i)
Equivalent series resistance equals the sum of individual resistances.
The same current passes through a single unbranched path and voltage drops add.
- R_i
- resistance of element i (ohm)
Use when — The shared node between consecutive elements has no other branch.
Common trap — Calling elements series because they look consecutive in the drawing.
1/R_eq = sum(1/R_i)
The reciprocal of the equivalent parallel resistance equals the sum of the reciprocals of the branch resistances.
Each branch has the same potential difference and branch currents add.
- R_i
- resistance of branch i (ohm)
Use when — Every component connects across the same two nodes.
Common trap — Forgetting that equivalent resistance is smaller than the smallest positive branch resistance.
V_1 = V R_1/(R_1 + R_2) ; I_1 = I R_2/(R_1 + R_2)
Voltage across the first resistor equals total voltage times its resistance over the sum; current in the first branch equals total current times the other resistance over the sum.
Series voltage divides in proportion to resistance; parallel current divides inversely with resistance.
- V_1
- voltage across R_1 (V)
- I_1
- current through R_1 (A)
Use when — The network is the stated unloaded series divider or a simple two-branch parallel divider.
Common trap — Using the voltage-divider result after a load changes the network.
P = VI ; Ohmic resistor: P = I^2 R = V^2/R ; W = VIt = Pt
Power equals voltage times current; for an Ohmic resistor it also equals current squared times resistance, or voltage squared divided by resistance.
Electrical energy transferred per unit time.
- P
- power (W)
- W
- energy (J)
- t
- time (s)
Use when — The voltage and current refer to the same element with a clear sign convention.
Common trap — Using V^2/R for a non-Ohmic device, or using the total supply voltage across only one resistor.
discharging: V = epsilon - Ir ; charging: V = epsilon + Ir ; I = epsilon/(R + r)
For a discharging cell, terminal voltage equals emf minus current times internal resistance; for a charging cell it equals emf plus that internal drop.
Emf is source energy per unit charge; terminal voltage accounts for internal potential change while current flows.
- epsilon
- emf (V)
- V
- terminal voltage (V)
- r
- internal resistance (ohm)
- R
- external load (ohm)
Use when — Current direction relative to cell polarity is known.
Common trap — Calling emf a force, or using the discharging sign for a charging cell.
identical aiding cells: epsilon_eq = n epsilon ; r_eq = n r ; I = n epsilon/(R + n r)
For n identical aiding cells in series, equivalent emf is n times emf and equivalent internal resistance is n times r.
Series connection adds signed emfs and internal resistances.
- n
- number of identical cells
- epsilon
- emf of one cell (V)
- r
- internal resistance of one cell (ohm)
Use when — Cell orientation is identified.
Common trap — Adding emf magnitudes when one cell opposes another.
identical cells: epsilon_eq = epsilon, r_eq = r/n ; unequal: r_eq = 1/sum(1/r_i), epsilon_eq = sum(epsilon_i/r_i) / sum(1/r_i)
For identical parallel cells, equivalent emf is unchanged and equivalent internal resistance is r over n; for unequal cells, weight each emf by the reciprocal of its internal resistance.
Parallel cells reduce equivalent internal resistance; unequal cells require internal-resistance weighting.
- epsilon_i
- emf of branch i (V)
- r_i
- internal resistance of branch i (ohm)
Use when — Polarities and internal resistances are included and the source model is valid.
Common trap — Averaging unequal emfs arithmetically without considering internal resistance.
sum(I) = 0 at a node
The algebraic sum of currents at a node is zero.
Steady-state charge conservation: total current entering equals total current leaving.
- I
- signed branch current at the node (A)
Use when — Branches meet at a node.
Common trap — Writing a separate independent junction equation for every node when one is redundant.
sum(Delta V) = 0 around a closed loop ; resistor with current: -IR ; against current: +IR ; cell minus to plus: +epsilon ; plus to minus: -epsilon
The algebraic sum of potential changes around a closed loop is zero, using one consistent sign convention for resistors and cells.
Energy-per-charge balance around a closed loop.
- Delta V
- signed potential change across an element (V)
Use when — A network cannot be fully solved by direct reduction or balance.
Common trap — Changing sign rules between terms, or treating a negative solved current as invalid.
P/Q = R/S at balance
At Wheatstone balance, the ratio of P to Q equals the ratio of R to S.
At balance the two detector nodes are equipotential and galvanometer current is zero.
- P, Q, R, S
- bridge arm resistances as labelled in the diagram (ohm)
Use when — A null condition is stated or established.
Common trap — Applying the balance ratio to an unbalanced bridge.
X/R = l/(100 - l) for X in the left gap and l measured from the left end
The unknown resistance divided by the known resistance equals the balance length divided by one hundred minus that length, for the stated placement.
Uniform-wire resistance is proportional to length, so the bridge ratio becomes a length ratio.
- X
- unknown resistance in the left gap (ohm)
- R
- known resistance in the right gap (ohm)
- l
- balance length from the left end, between 0 and 100 (cm)
Use when — The bridge wire is uniform and the null point is measured from the stated end.
Common trap — Copying the ratio without checking gap placement, measurement direction, end resistance and whether the null point exists.
