Eight topics on what actually moves around a loop and what gets spent doing it. Current as a rate that is conserved along a path, simple circuits and what a complete path requires, resistance built from a material property plus a shape, power matched to the element that owns every symbol in the formula, compound networks read from the nodes rather than the drawing, the loop and junction rules as energy and charge conservation, and RC circuits, where the answer depends on when you look.
I = Δq / Δt, in amperes. The SAME everywhere along an unbroken series path: nothing consumes charge
What a bulb spends
energy per charge, which shows up as a potential difference across it. Two separate ledgers: charge per second, and joules per coulomb
What a battery fixes
the terminal potential difference, NOT the current. Change the network and solve for I again
Ohm's law
I = ΔV / R, for OHMIC materials, where R holds constant as I varies. A hot filament is not one
Resistance of an object
R = ρL/A. ρ belongs to the MATERIAL; L and A turn it into the resistance of a particular wire
Wire geometry
A = πr², so doubling the radius QUARTERS R. Length scales R directly
Reading a graph
I against ΔV: slope = 1/R, so the steeper line is the SMALLER resistance. Axes swapped: slope = R
Power
P = IΔV = I²R = (ΔV)²/R. Every symbol must belong to the SAME element
Which form ranks brightness
shared current (series): P = I²R favours the LARGER R. Shared ΔV (parallel): P = (ΔV)²/R favours the SMALLER R
Energy from power
E = PΔt. A watt already contains the per second, so a duration MULTIPLIES. 1 kWh = 3.6 × 10⁶ J
Real battery
terminal ΔV = ε − Ir. Equals ε only at zero current; a plot against I has slope −r
Series or parallel
decided by NODES, not by the drawing. Same two nodes = parallel; one shared node nothing else touches = series
Resistors in series
R(eq) = R₁ + R₂ + ... Same current through all; ΔV divides in PROPORTION to R
Resistors in parallel
1/R(eq) = 1/R₁ + 1/R₂ + ... Remember the final flip. R(eq) is BELOW every branch
Adding a parallel branch
adds a PATH: R(eq) falls and the battery current RISES. Only a series addition adds opposition
Short circuit
a zero-resistance path across an element puts both its ends at one potential, so its current is ZERO
Meters
ammeter IN the path, ideally 0 Ω. Voltmeter ACROSS two points, ideally infinite Ω. Swapped, each destroys the circuit it measures
Loop rule
ΣΔV = 0 around ANY closed path, source or no source. Signs come from the direction you walk, not from the component type
Junction rule
current in = current out. It fixes the TOTAL; the shared ΔV and each branch's R fix the shares
Capacitors in parallel
C(eq) = C₁ + C₂ + ... OPPOSITE to resistors
Capacitors in series
1/C(eq) = 1/C₁ + 1/C₂ + ... Below the smallest; each carries the SAME magnitude of charge
RC at t = 0
an UNCHARGED capacitor acts like a plain wire: replace it with one and solve. Current is at its largest
RC after a long time
no current in that branch: open it and solve. The capacitor reads the ΔV across the two points it bridges, often not ε
Time constant
τ = RC. At t = τ charging is ~63% done, discharging is at ~37%. Larger R or C means SLOWER. Allow ~5τ
Unit 11 tools
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