Chapter 13: Current Electricity

• Electric Current (I)

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  • Rate of flow of charge.
  • $I = \frac{\Delta Q}{\Delta t}$.
  • Unit: Ampere (A) = C/s.
  • Dimensions: $[A]$.
  • Scalar quantity (though direction is associated, it doesn't follow vector addition rules).
  • Conventional current: direction of positive charge flow.
  • Electron flow: opposite to conventional current.

  • Current Density (J)

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    • Current per unit cross-sectional area.
    • $J = \frac{I}{A}$.
    • Unit: $A/m^2$.
    • Vector quantity.

  • Drift Velocity ($v_d$)

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    • Average velocity of charge carriers in a conductor due to electric field.
    • $I = n A v_d q$.
    • $n$: number density of charge carriers.
    • $A$: cross-sectional area.
    • $q$: charge of each carrier.
    • $v_d$ is typically very small.

• Ohm's Law

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  • Current through a conductor is directly proportional to the potential difference across its ends, provided temperature and other physical conditions remain constant.
  • $V = IR$.
  • $V$: potential difference.
  • $I$: current.
  • $R$: resistance.

  • Resistance (R)

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    • Opposition to the flow of current.
    • Unit: Ohm ($\Omega$) = V/A.
    • Dimensions: $[ML^2T^{-3}A^{-2}]$.
    • Depends on:
      • Nature of material (resistivity).
      • Length of conductor ($R \propto L$).
      • Cross-sectional area ($R \propto \frac{1}{A}$).
      • Temperature.
    • $R = \rho \frac{L}{A}$.
    • $\rho$: resistivity.

  • Resistivity ($\rho$)

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    • Intrinsic property of a material.
    • Unit: Ohm-meter ($\Omega~m$).
    • Temperature dependence: $\rho_T = \rho_o (1 + \alpha \Delta T)$.
    • $\alpha$: temperature coefficient of resistivity.

  • Conductance (G)

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    • Reciprocal of resistance.
    • $G = \frac{1}{R}$.
    • Unit: Siemens (S) or mho ($\Omega^{-1}$).

  • Conductivity ($\sigma$)

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    • Reciprocal of resistivity.
    • $\sigma = \frac{1}{\rho}$.
    • Unit: Siemens per meter ($S/m$).

• Combinations of Resistors

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  • Series Combination

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    • Current is same through each resistor.
    • Total voltage: $V = V_1 + V_2 + ...$.
    • Equivalent resistance: $R_{eq} = R_1 + R_2 + ...$.
    • $R_{eq}$ is always greater than the largest individual resistance.

  • Parallel Combination

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    • Voltage is same across each resistor.
    • Total current: $I = I_1 + I_2 + ...$.
    • Equivalent resistance: $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + ...$.
    • For two resistors: $R_{eq} = \frac{R_1 R_2}{R_1 + R_2}$.
    • $R_{eq}$ is always less than the smallest individual resistance.

• Joule's Law (Heating Effect of Current)

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  • Heat produced in a resistor is proportional to the square of current, resistance, and time.
  • $H = I^2 R t$.
  • Also: $H = V I t = \frac{V^2}{R} t$.
  • Unit: Joule (J).

• Electric Power (P)

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  • Rate at which electrical energy is converted into other forms of energy.
  • $P = V I = I^2 R = \frac{V^2}{R}$.
  • Unit: Watt (W).

• Electromotive Force (EMF) and Terminal Potential Difference (TPD)

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  • Electromotive Force (EMF, $\mathcal{E}$)

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    • Energy supplied by a source per unit charge to drive current around a circuit.
    • Maximum potential difference across the terminals of a source when no current is drawn.
    • Unit: Volt (V).

  • Terminal Potential Difference (TPD, V)

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    • Potential difference across the terminals of a source when current is drawn.
    • $V = \mathcal{E} - Ir$.
    • $r$: internal resistance of the source.
    • If $I=0$, then $V = \mathcal{E}$.

  • Internal Resistance (r)

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    • Resistance offered by the source itself to the flow of current.
    • Causes a voltage drop within the source.
    • $r = \frac{\mathcal{E} - V}{I}$.

• Kirchhoff's Rules

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  • Used to analyze complex circuits.

  • Kirchhoff's First Rule (Current Rule / Junction Rule)

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    • Sum of currents entering a junction equals sum of currents leaving it.
    • $\sum I_{in} = \sum I_{out}$.
    • Based on conservation of charge.

  • Kirchhoff's Second Rule (Voltage Rule / Loop Rule)

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    • Algebraic sum of potential changes around any closed loop is zero.
    • $\sum \Delta V = 0$.
    • Based on conservation of energy.

• Wheatstone Bridge

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  • Circuit used to measure an unknown resistance accurately.
  • Balanced condition: $\frac{R_1}{R_2} = \frac{R_3}{R_4}$.
  • No current flows through the galvanometer in balanced condition.

• Potentiometer

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  • Device used to measure potential difference or EMF accurately.
  • Works on the principle that potential drop across a wire is proportional to its length (for uniform wire and constant current).
  • Can compare EMFs of two cells.

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