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 an electric field within the conductor, typically established by a voltage source.

• Electric Current (I): Defined as the rate of flow of charge, measured in Amperes (A).

• Formula: $I = \frac{\Delta Q}{\Delta t}$ where $\Delta Q$ is the charge in Coulombs and $\Delta t$ is the time in seconds.

• Conventional Current: Treated as the 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

The flow of charge in a conductor is uniform due to the presence of an internal electric field. At non-zero temperature, electrons move randomly, but the electric field causes a net drift.

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

• Random Motion: Without an electric field, electrons move randomly and do not contribute to net current.

• With Electric Field: Electrons acquire a drift velocity along the direction opposite to the field.

• Example: When a circuit is switched on, the electric field propagates quickly, causing immediate current flow.

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 a linear fashion for many materials.

• Ohm's Law: $V = IR$ where $V$ is voltage (Volts), $I$ is current (Amperes), and $R$ is resistance (Ohms).

• Ohmic Resistor: A resistor that obeys Ohm's Law, showing a linear relationship between $V$ and $I$.

• Non-Ohmic Resistor: Devices like diodes do not show a linear $V$-$I$ relationship.

• Example: A metal wire at room temperature typically behaves as an Ohmic resistor.

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 ($\Omega \cdot m$), $L$ is length (m), and $A$ is cross-sectional area ($m^2$).

• Dependence:

  • Increases with length ($L$)

  • Decreases with area ($A$)

  • Depends on material (resistivity $\rho$)

• Resistivity Table:

  Substance	Resistivity $\rho$ ($\Omega \cdot m$)
  Mercury	9.5 × 10-7
  Nickrome alloy	1.10 × 10-6
  Glass	1 × 1010
  Wood	1 × 1010
  Additional info: Resistivity increases with temperature for metals.

• Example: Doubling the length of a wire doubles its resistance; doubling the area halves its resistance.

Applications of Ohm's Law

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

• Example Calculation: If $V = 24$ V and $I = 0.24$ A, then $R = \frac{V}{I} = 100 \Omega$.

• Changing Material: If resistivity doubles, current halves for the same voltage and dimensions.

• Series Connection: Two rods of resistance $R$ connected end-to-end have total resistance $2R$.

Summary of Key Units

• Charge: Coulombs (C)

• Current: Amperes (A = C/s)

• Potential: Volts (V)

• Energy: Joules (J = CV)

• Power: Watts (W = J/s)

• Resistance: Ohms ($\Omega$ = V/A)

Additional info: These notes cover the foundational concepts of current, resistance, and DC circuits, including microscopic and macroscopic views, Ohm's Law, and practical applications in circuit analysis.