Closure Audits

Capacitance Inversion Audit

read Q = C V three ways

The capacitor equation is solved forward and backward so charge, voltage, and capacitance are seen as one exact unit ledger.

capacitors Use three scan tables to read Q = C V as charge, voltage, and capacitance.

Example

The capacitor equation is solved forward and backward so charge, voltage, and capacitance are seen as one exact unit ledger. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

One capacitor equation can be read three ways

The same ideal-capacitor relation can solve for charge, voltage, or capacitance. The middle checked case uses 3 farads and 4 volts, which gives 12 coulombs.

Q=CV,V=QC,C=QVQ = C V,\qquad V=\frac{Q}{C},\qquad C=\frac{Q}{V}
Checked capacitor rowThe diagram capacitance is the middle row used in the scans.3 F

Forward reading: capacitance times voltage gives charge

Read the first table left to right. Each row multiplies capacitance by voltage to close the charge column.

CVQ2 F3 V6 C3 F4 V12 C4 F5 V20 C\begin{array}{c|c|c}C&V&Q\\2\ \text{F}&3\ \text{V}&6\ \text{C}\\3\ \text{F}&4\ \text{V}&12\ \text{C}\\4\ \text{F}&5\ \text{V}&20\ \text{C}\\\end{array}
Forward charge scanThe middle table row is the checked capacitor.3 F

Inverse reading: charge divided by capacitance gives voltage

Now hold charge at 12 coulombs and divide by three different capacitances. The same equation is being read from charge back to voltage.

QCV12 C2 F6 V12 C3 F4 V12 C4 F3 V\begin{array}{c|c|c}Q&C&V\\12\ \text{C}&2\ \text{F}&6\ \text{V}\\12\ \text{C}&3\ \text{F}&4\ \text{V}\\12\ \text{C}&4\ \text{F}&3\ \text{V}\\\end{array}
Voltage inverse scanThe middle row again matches the checked capacitor.3 F

Inverse reading: charge divided by voltage gives capacitance

Finally divide charge by voltage. Three rows with matching ratios all recover the same 3 farad capacitance.

QVC12 C4 V3 F18 C6 V3 F24 C8 V3 F\begin{array}{c|c|c}Q&V&C\\12\ \text{C}&4\ \text{V}&3\ \text{F}\\18\ \text{C}&6\ \text{V}&3\ \text{F}\\24\ \text{C}&8\ \text{V}&3\ \text{F}\\\end{array}
Capacitance inverse scanThe middle row has the same capacitance as the diagram.3 F

The unit algebra closes the three readings

Coulombs per volt returns farads, and coulombs per farad returns volts. This is why the three table readings are one equation, not three separate rules.

CV=FCF=VF⋅V=C\frac{\text{C}}{\text{V}}=\text{F}\qquad \frac{\text{C}}{\text{F}}=\text{V}\qquad \text{F}\cdot\text{V}=\text{C}
Checked capacitor rowThe same checked capacitor supports each reading.3 F