Circuits

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Table of Contents

• Circuits
  • Key Definitions
  • Steady-State Circuits
  • Potential in a Circuit
    • $V_{source}=IR$
  • Energy and Power
    • $P=IV$
    • $P=IV=I^{2}R=\frac{V^2}{R}$
  • Circuit Organization
    • In Series
      • Potential Difference
        • $\mathrm{\Delta }{V}_{eq}=\sum \mathrm{\Delta }{V}_{R_i}$
      • Current
        • $I_{eq}=I_{R_i}$
      • Resistance
        • $R_{eq}=\sum R_i$
      • Capacitance
        • $\frac{1}{C_{eq}}=\sum \frac{1}{C_i}$
    • In Parallel
      • Potential Difference
        • $\mathrm{\Delta }{V}_{eq}=\mathrm{\Delta }{V}_{R_i}$
      • Current
        • $I_{eq}=\sum I_{R_i}$
      • Resistance
        • $\frac{1}{R_{eq}}=\sum \frac{1}{R_i}$
      • Capacitance
        • $C_{eq}=\sum C_i$
  • Kirchhoff’s Rules
    • Current/Junction Rule
      • $\sum I_{in}=\sum I_{out}$
    • Voltage/Loop Rule
      • $\oint \overrightarrow{E}\cdot d\overrightarrow{l}=0$
      • $(\sum \mathrm{\Delta }V{)}_{loop}=0$

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Circuits

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Key Definitions

Steady-state circuit - circuit which maintains constant current

Power - the rate of energy delivered or extracted from a circuit

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Steady-State Circuits

• Conductors must form a closed loop to maintain a steady
• Additionally, a source of constant voltage must be in the circuit (unless using a superconductor)
• Steady current requires the to be constant and thus the to be constant

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Potential in a Circuit

The net change in potential energy for a travelling around a circuit must be zero Thus, the electric potential around a circuit must also be 0. I.e.

$V_{source}=IR$

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Energy and Power

The power flowing through a circuit is given by:

$P=IV$

The power is independent of Ohm’s Law, but if the law is true, then:

$P=IV=I^{2}R=\frac{V^2}{R}$

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Circuit Organization

In Series

Potential Difference

$\mathrm{\Delta }{V}_{eq}=\sum \mathrm{\Delta }{V}_{R_i}$

Current

$I_{eq}=I_{R_i}$

Resistance

$R_{eq}=\sum R_i$

Capacitance

$\frac{1}{C_{eq}}=\sum \frac{1}{C_i}$

In Parallel

Potential Difference

$\mathrm{\Delta }{V}_{eq}=\mathrm{\Delta }{V}_{R_i}$

Current

$I_{eq}=\sum I_{R_i}$

Resistance

$\frac{1}{R_{eq}}=\sum \frac{1}{R_i}$

Capacitance

$C_{eq}=\sum C_i$

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Kirchhoff’s Rules

Current/Junction Rule

The current entering a junction is equivalent to the current leaving the junction:

$\sum I_{in}=\sum I_{out}$

Voltage/Loop Rule

Because in a DC circuit the draining the current in the circuit is conservative:

$\oint \overrightarrow{E}\cdot d\overrightarrow{l}=0$

Thus in closed loop circuits:

$(\sum \mathrm{\Delta }V{)}_{loop}=0$