Comprehensive Notes – Capacitors, Current, Resistivity & Ohm’s Law

Capacitors in Series: Core Rules

  • When capacitors share a single conducting path (no junctions), they are in series.
  • Charge movement chain:
    • Charge +q is forced onto the first plate of C_1.
    • The same magnitude charge -q is induced on the opposite plate, which in turn becomes +q on the next capacitor, and so on.
    • RESULT: All series capacitors carry identical charge q{1}=q{2}=\dots=q{n}=q{eq}.
  • Potential (voltage) drops distribute:
    • Total battery voltage equals sum of individual drops.
    • \Delta V{eq}=\sum{i=1}^{n}\Delta V_i
  • Equivalent capacitance for capacitors in series:
    \frac{1}{C{eq}}=\sum{i=1}^{n}\frac{1}{Ci}\quad\Rightarrow\quad C{eq}=\Big(\sum \frac{1}{C_i}\Big)^{-1}
  • Physical analogy: Placing two identical plate capacitors in series is like making one larger capacitor with double the separation d, hence lower C (because C\propto 1/d).

Worked Example – 3 µF, 6 µF, 12 µF, 24 µF in Series (Battery 18 V)

  • Identify: single loop → series.
  • Compute C{eq}: \frac{1}{C{eq}}=3^{-1}+6^{-1}+12^{-1}+24^{-1}=0.331\;\mu F^{-1} C_{eq}=\frac{1}{0.625}=1.6\;\mu F (calculator showed 1/0.625 = 1.6)
    • Check: 1.6 µF is smaller than each original value → consistency.
  • Charge on the equivalent capacitor: q=C_{eq}V=(1.6\,\mu F)(18\,\text{V})=28.8\,\mu C
    • Series rule → every capacitor gets 28.8\,\mu C.
  • Individual voltages using V=q/C:
    • V_1=\frac{28.8}{3}=9.6\,\text{V}
    • V_2=\frac{28.8}{6}=4.8\,\text{V}
    • V_3=\frac{28.8}{12}=2.4\,\text{V}
    • V_4=\frac{28.8}{24}=1.2\,\text{V}
    • Check sum: 9.6+4.8+2.4+1.2=18\,\text{V} ✓
  • Qualitative takeaway: smaller C → larger V in series such that q=CV holds for each.

Mixed Series–Parallel Example (4 µF, 2 µF, 1.5 µF & 12 V Battery)

  • Step-by-step reduction stressed (pause and redraw method).
  1. First identification: 2 µF and 1.5 µF share both nodes → parallel.
    • C_{(2\parallel1.5)} = 2+1.5 = 3.5\,\mu F
  2. New circuit: 4 µF series with 3.5 µF.
    • \frac{1}{C{eq}}=4^{-1}+3.5^{-1}\Rightarrow C{eq}=1.867\,\mu F\approx1.9\,\mu F (2 sig-fig)

Charges & Voltages (Back-substitution)

  • Equivalent charge:
    q=C_{eq}V=(1.867\,\mu F)(12\,\text{V})=22.4\,\mu C
  • Series part (4 µF vs combined 3.5 µF): share the same charge 22.4 µC.
    • V_{4\,\mu F}=\frac{22.4}{4}=5.6\,\text{V}
    • Remaining voltage on parallel combo:
      V_{3.5\,\mu F}=12-5.6=6.4\,\text{V}
  • Parallel rule → both inner capacitors have 6.4 V.
    • q_{2\,\mu F}=2\,\mu F\times6.4\,\text{V}=12.8\,\mu C
    • q_{1.5\,\mu F}=1.5\,\mu F\times6.4\,\text{V}=9.6\,\mu C
    • Check: 12.8+9.6=22.4\,\mu C ✓
  • Final answers:
    • Charges: 22.4 µC (4 µF), 12.8 µC (2 µF), 9.6 µC (1.5 µF).
    • Voltages: 5.6 V, 6.4 V, 6.4 V respectively.
  • Pedagogical note: constantly verify with conservation rules (voltages in series sum, charges in parallel sum).

Energy Stored in a Capacitor

  • Physical process: moving incremental dq from one plate to the other through potential V(q).
  • Work element: dU = V(q)\,dq.
  • Integrate (linear V=\frac{q}{C}) from 0→Q:
    U = \int_0^{Q} \frac{q}{C}\,dq = \frac{1}{2}\frac{Q^2}{C}
  • Alternate interchangeable forms via Q=CV:
    U=\frac{1}{2}Q V = \frac{1}{2}C V^2 = \frac{1}{2}\frac{Q^2}{C}
  • Graphical justification: area under a straight-line V(q) plot ⇒ right triangle \frac{1}{2}QV.
  • Practical insight: only half the battery’s chemical work becomes stored electrical energy; the other half is dissipated (usually as heat) during charging.

Introduction to Electric Current

  • Definition: rate of charge passage through a cross-section.
    I=\frac{\Delta Q}{\Delta t}
  • Unit: ampere (A) = \text{Coulomb}\,/\,\text{second}.
  • Quick example: Electric eel moves 2\,\text{mC} in 2\,\text{ms} ⇒ I=1\,\text{A}.
  • Empirical indicators of current:
    • Wire warms.
    • Incandescent bulb glows; brightness ∝ current.
    • Compass needle deflects (magnetic field from current).
  • Direction convention: Conventional current = flow of positive charge (opposite electron motion).

Conservation of Current (Kirchhoff Junction Rule)

  • At any junction: \sum I{in}=\sum I{out}.
  • Consequence of charge conservation (no creation/annihilation in wires).

Microscopic Picture: Drift vs Signal

  • Electrons undergo frequent collisions with lattice ions; average drift velocity very small \approx10^{-4}\,\text{m/s}.
  • Electric field acceleration: a=\frac{eE}{me}, with average time between collisions \tau ⇒ vd=\frac{eE\tau}{2m_e}.
  • Electrical signal (field change) propagates near 10^{8}\,\text{m/s} ((\sim c/3)); akin to incompressible water shove.
  • Water-hose analogy: pump start moves water everywhere almost simultaneously even though molecules drift slowly.

Resistivity \rho – Material Property

  • Defines how strongly a material opposes current independent of geometry.
  • Units: \Omega\,m.
  • Categories (order-of-magnitude):
    • Conductors: \rho\sim10^{-8}\,\Omega m (Cu, Ag).
    • Semiconductors: 10^{-5}!\text{–}!10^{-3}\,\Omega m (Si, Ge); \rho drops when heated (negative \alpha).
    • Insulators: 10^{8}\,\Omega m and up (glass, rubber).
  • Temperature dependence: \rho(T)=\rho0[1+\alpha (T-T0)]
    • \alpha = temperature coefficient (positive for metals, negative for semiconductors).

Resistance R of a Conductor

  • Geometry + material: R=\rho\,\frac{L}{A}
    • L = length (m)
    • A = cross-section (m^2). For circular wire A=\pi r^2=\pi d^2/4.
  • Temperature also: R(T)=R0[1+\alpha (T-T0)].
  • Units: ohm (Ω) where 1\,\Omega=1\,\text{V}/\text{A}.

Batteries, EMF & Ohm’s Law Revisited

  • Battery maintains nearly constant terminal voltage (EMF \mathcal{E}) until near depletion.
    \mathcal{E}=\frac{W_{chem}}{q}
  • Ohm’s law for ohmic materials: \Delta V = I R
    • Think independent ((R,\Delta V)) vs dependent ((I)).
  • Changing parameters:
    • Double \Delta V with same R → double I.
    • Thicker/shorter wire (lower R) at same \Delta V → higher I.
    • Material switch (Cu→Fe) raises \rho hence lowers I.

Ohmic vs Non-Ohmic Devices

  • Ohmic: constant R over operating range (standard resistors, many metals at modest temperatures).
  • Non-ohmic examples:
    • Incandescent bulb (tungsten R rises steeply with temperature → curved V–I graph).
    • Batteries (supply fixed \Delta V not fixed R).
    • Capacitors/diodes etc. (current depends non-linearly on voltage or time).

Example – 100 W Incandescent Bulb (Tungsten)

  • Data: P=100\,\text{W},\;I=0.83\,\text{A},\;V=120\,\text{V}.
  • (a) Filament resistance:
    R=\frac{V}{I}=\frac{120}{0.83}=1.45\times10^{2}\,\Omega
  • (b) Length of 35 µm-diameter tungsten wire operating at ~1500 °C.
    • Resistivity at 1500 °C: \rho\approx5\times10^{-7}\,\Omega m.
    • Area: A=\pi(d/2)^2=\pi (0.035\times10^{-3})^2/4=9.62\times10^{-10}\,m^2.
    • L=R\,A/\rho=145\,\Omega\,\cdot9.62\times10^{-10}/(5\times10^{-7})\approx0.278\,m\,(28\,cm\;\approx11\,in).
    • Explains tightly coiled long filament inside bulb.

Conceptual & Real-World Connections

  • Capacitor energy math links directly to kinetic → potential energy transfer concept; half energy lost observable as heat.
  • Temperature coefficients exploited:
    • Tungsten’s large \alpha critical for bulbs that self-heat to incandescence.
    • Negative \alpha of semiconductors enables thermistors and temperature sensors.
  • Magnetic effects of current motivate later study (Ch. 11): compass deflection demonstration.
  • Practical advice for circuit problems: redraw after every simplification; annotate Q, \Delta V at each stage; constantly check conservation rules.

Quick Reference: Key Equations

  • Series C: \frac{1}{C{eq}}=\sum\frac{1}{Ci}
  • Parallel C: C{eq}=\sum Ci
  • Capacitor energy: U=\frac{1}{2}CV^2=\frac{1}{2}QV
  • Current: I=\dfrac{\Delta Q}{\Delta t}
  • Junction rule: \sum I{in}=\sum I{out}
  • Resistivity: \rho(T)=\rho_0[1+\alpha\Delta T]
  • Resistance: R=\rho\dfrac{L}{A}, R(T)=R_0[1+\alpha\Delta T]
  • Ohm’s law: V=IR
  • Power (preview): P=IV=I^2R=\dfrac{V^2}{R} (foundation for example, detailed later).

These notes capture all primary derivations, numeric illustrations, conceptual analogies, and practical strategies discussed in the transcript for exam preparation.