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Physics: Electric Circuits and Capacitors

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  • Equivalent resistance of resistors in series
    The equivalent resistance is the sum: \(R_{eq}=R_1+R_2+R_3+\dots\).
  • Current in resistors connected in series
    The current is the same through all resistors: \(I_{tot}=I_1=I_2=I_3=\dots\).
  • Voltage drops across resistors in series
    The total voltage drop is the sum of individual drops: \(V_{tot}=V_1+V_2+V_3+\dots\).
  • Equivalent resistance of resistors in parallel
    The reciprocal of equivalent resistance is the sum of reciprocals: \(\frac{1}{R_{eq}}=\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3}+\dots\).
  • Current in resistors connected in parallel
    The total current is the sum of currents through each resistor: \(I_{tot}=I_1+I_2+I_3+\dots\).
  • Voltage across resistors in parallel
    The voltage across each resistor is the same: \(V_{tot}=V_1=V_2=V_3=\dots\).
  • Kirchhoff's Junction Rule
    The sum of currents entering a junction equals zero: \(\sum I_i=0\) (currents into junction positive, out negative).
  • Kirchhoff's Loop Rule
    The sum of potential changes around any closed loop is zero: \(\sum \Delta V_i=0\).
  • Potential change moving in direction of current over resistor
    Potential drop is negative: \(\Delta V = -RI\).
  • Potential change moving against direction of current over resistor
    Potential change is positive: \(\Delta V = +RI\).
  • Potential change moving from negative to positive terminal of battery
    Potential change is positive: \(\Delta V = +E\).
  • Potential change moving from positive to negative terminal of battery
    Potential change is negative: \(\Delta V = -E\).
  • Time constant of an RC circuit
    The time constant is \(\tau=RC\), representing the characteristic time for charging or discharging.
  • Charging current in an RC circuit
    Current decreases exponentially: \(i(t)=I_0 e^{-t/\tau}\), where \(I_0=\frac{\varepsilon}{R}\).
  • Charging charge on capacitor in RC circuit
    Charge increases as: \(q(t)=Q_0(1-e^{-t/\tau})\), with \(Q_0=\varepsilon C\).
  • Discharging charge on capacitor in RC circuit
    Charge decreases exponentially: \(q(t)=Q_0 e^{-t/\tau}\).
  • Definition of capacitance
    Capacitance relates charge and voltage: \(Q=VC\).
  • Equivalent capacitance of capacitors in series
    Reciprocal of equivalent capacitance is sum of reciprocals: \(\frac{1}{C_{eq}}=\frac{1}{C_1}+\frac{1}{C_2}+\frac{1}{C_3}+\dots\).
  • Equivalent capacitance of capacitors in parallel
    Equivalent capacitance is sum: \(C_{eq}=C_1+C_2+C_3+\dots\).
  • Potential energy stored in a capacitor
    Energy stored is \(U_C=\frac{1}{2}CV^2 = \frac{Q^2}{2C} = \frac{1}{2}QV\).