Current, Resistance, and DC Circuits – Study Notes

Current, Resistance, and DC Circuits

Introduction to Electric Current

Electric current is a fundamental concept in physics, describing the flow of electric charge through a conductor. In direct current (DC) circuits, this flow is maintained by a constant electric field.

• Electric Current (I): Defined as the rate of flow of charge through a surface. The SI unit is the ampere (A).

• Formula: $I = \frac{\Delta Q}{\Delta t}$ where $\Delta Q$ is the charge passing through a cross-section in time $\Delta t$.

• Conventional Current: Treated as a flow of positive charges, regardless of the actual charge carriers (which are often electrons).

• Direction: In metallic conductors, electrons move opposite to the direction of conventional current.

• Example: In a copper wire, electrons drift from the negative to the positive terminal, but current is defined from positive to negative.

Microscopic View of Current

At the atomic level, the flow of charge is uniform throughout a conductor. In the absence of an electric field, electrons move randomly; with an applied field, they acquire a net drift velocity.

• Drift Velocity: The average velocity of charge carriers due to an electric field.

• Random Motion: At non-zero temperature, electrons move randomly even without an applied field.

• Effect of Electric Field: The field causes a net displacement of electrons, resulting in current.

• Example: The drift velocity in metals is typically $\sim 10^{-4}$ m/s, while the electric field propagates at nearly the speed of light.

Resistance and Ohm's Law

Resistance quantifies how much a material opposes the flow of electric current. Ohm's Law relates current, voltage, and resistance in many materials.

• Resistance (R): The property of a conductor that resists the flow of current. Measured in ohms (\Omega).

• Ohm's Law: $V = IR$ where $V$ is the potential difference, $I$ is the current, and $R$ is the resistance.

• Ohmic vs. Non-Ohmic Materials:

  • Ohmic: Current is proportional to voltage (e.g., metal wires).

  • Non-Ohmic: Current does not vary linearly with voltage (e.g., diodes).

• Example: For a resistor with $V = 24$ V and $I = 0.24$ A, $R = \frac{24}{0.24} = 100 \Omega$.

Resistance of a Wire

The resistance of a cylindrical wire depends on its length, cross-sectional area, and the material's resistivity.

• Formula: $R = \rho \frac{L}{A}$ where $\rho$ is resistivity, $L$ is length, and $A$ is cross-sectional area.

• Dependence:

  • Increases with length ($L$)

  • Decreases with area ($A$)

  • Depends on material (resistivity $\rho$)

• Resistivity Table:

  Substance	$\rho$ ($\Omega \cdot$ m)
  Mercury	$9.5 \times 10^{-7}$
  Nickrome alloy	$1.10 \times 10^{-6}$
  Glass	$1 \times 10^{10}$
  Wood	$1 \times 10^{10}$
  Additional info: Table includes other materials such as copper, silver, and insulators.

• Temperature Dependence: For metals, resistivity increases with temperature.

• Example: Two rods of length $L$ welded end to end have resistance $2R$.

Applications of Ohm's Law

Ohm's Law is used to solve for unknown quantities in circuits, such as current, voltage, or resistance.

• Example Calculation:

  • Given $V = 24$ V, $R = 100 \Omega$, $I = \frac{V}{R} = 0.24$ A.

  • If resistivity doubles, $R$ doubles, and $I$ halves.

• Series Connection:

  • Resistances add: $R_{\text{total}} = R_1 + R_2 + ...$

  • Current is the same through all components.

• Parallel Connection:

  • Reciprocal of resistances add: $\frac{1}{R_{\text{total}}} = \frac{1}{R_1} + \frac{1}{R_2} + ...$

  • Voltage is the same across all branches.

Summary of Units

Quantity	Unit	Symbol
Charge	Coulomb	C
Current	Ampere	A
Potential	Volt	V
Energy	Joule	J
Power	Watt	W
Resistance	Ohm	\Omega

Additional info:

• These notes are based on lecture slides and textbook-style explanations, with some inferred context for completeness.

• Further topics such as electromotive force (emf), measurement devices, and Kirchhoff's rules are covered in subsequent sections of the chapter.