19. Capacitance

Capacitance C is the charge stored per volt: C=Q/V. It depends on geometry and dielectric, not on Q and V separately for a fixed capacitor.

For an isolated spherical conductor capacitance depends on radius; for parallel plates it depends on area, separation and dielectric.

A larger isolated sphere has greater capacitance because more charge can be stored for the same potential.

Capacitance is not the same as capacity in coulombs and is not automatically increased by raising voltage.

For a fixed capacitor, Q=CV. The slope of a Q–V graph is C.

Use farads, coulombs and volts; identify whether voltage is across the capacitor and account for dielectric or geometry changes before treating C as constant.

A 220 μF capacitor charged to 5.0 V stores 1.1×10⁻³ C.

The capacitor does not store a fixed amount of charge independent of voltage; C is the proportionality constant.

For capacitors in parallel, C_total=ΣC because voltage is common; in series, 1/C_total=Σ(1/C) because charge magnitude is common on each capacitor.

Draw the topology first and check bounds: parallel capacitance exceeds the largest branch; series capacitance is below the smallest.

Two equal capacitors C in parallel give 2C, while in series they give C/2.

Capacitors do not follow resistor combination rules automatically; the conserved/shared quantity is different.

For parallel capacitors C_total=ΣC; for series 1/C_total=Σ(1/C). Parallel capacitors share voltage, while series capacitors carry equal charge magnitude.

Draw the topology first and check limits: series C is below the smallest, parallel C above the largest.

Two equal 10 μF capacitors give 20 μF in parallel and 5 μF in series.

Capacitor combination rules differ from resistor rules because the shared/conserved quantity is different.

Energy stored in a capacitor is electric potential energy associated with separated charge and the electric field between conductors.

Charging requires work against the growing potential difference; discharging transfers this energy to other stores or radiation.

A camera flash capacitor releases stored electric energy rapidly through a lamp circuit.

The dielectric does not create energy; it changes capacitance and the energy relation for a given Q or V.

For a linear capacitor, stored energy W=½QV=½CV²=Q²/(2C). The half factor comes from the average voltage during charging.

Choose the form matching known Q, V or C and keep units in farads, volts and coulombs.

A 100 μF capacitor at 20 V stores 0.020 J.

Using QV without the half factor doubles the energy for a capacitor charged from zero.

A discharging capacitor has decreasing Q and V; current is related to the rate of charge change and initially has the greatest magnitude.

Use the circuit polarity and current direction consistently. The resistor receives energy as the capacitor’s electric field collapses.

A voltage-time trace falls steeply at first and then levels off, showing that the discharge rate slows.

Current is not constant during discharge, and a falling voltage does not mean the capacitor instantly becomes uncharged.

For a resistor R and capacitor C, τ=RC. After one time constant in a discharge, the relevant quantity has fallen to e⁻¹≈37% of its initial value.

Larger R or C makes the response slower. Identify the effective resistance seen by the capacitor, not every resistor in the diagram.

R=2.0 kΩ and C=100 μF gives τ=0.20 s.

τ is not the time to reach exactly zero; exponential decay approaches zero asymptotically.

A discharging quantity follows x=x₀e^(−t/RC); charging voltage follows V=V₀(1−e^(−t/RC)). Current and charge use the corresponding variable x.

At t=0 and after several τ, use limiting values to check the expression and identify whether the curve rises or falls.

After t=τ in a discharge, voltage and charge are 37% of their initial values; after about 5τ they are close to zero.

The exponent is dimensionless, so t and RC must have the same time units; exponential decay is not linear.