physics jan 14 2025

Resistance in Circuits

• Resistance Represented by 'r':

  • Resistance in electrical circuits is usually denoted by a lowercase 'r'.

  • Voltage from the battery (e.m.f) is reduced by the product of current (I) and internal resistance (r).

  • Formula: V = E - I * r

• Voltage and Electromotive Force (e.m.f):

  • The e.m.f from a battery can be expressed as:

    • E = I * (r + R)

  • Where 'r' is the internal resistance and 'R' is the external circuit resistance.

Example Problem

• Given Values:

  • E.m.f (E) = 24 volts

  • Resistance (R) = 23 ohms

  • Current (I) = 1 amp

• Analysis:

  • For ordinary calculations (Ohm's Law):

    • Expected voltage V = 24V / 23 ohms = approximately 1.04 amps (without considering internal resistance).

  • Internal Resistance Calculation:

    • Rework formula where E = I * (r + R):

      • 24 V = 1 A * (r + 23 ohms)

      • Therefore, 24 = 1 * (r + 23) implies:

        • r = 1 ohm

  • Internal resistance varies but is generally around 1 volt.

Non-Ideal Battery Characteristics

• Battery Imperfections:

  • Real batteries are not perfect; they produce heat due to their internal resistance.

  • Actual voltage available = E - I * r, indicating that as current (I) increases, the effective voltage can decrease.

  • In non-ideal batteries, increased current can lead to lower voltage in the circuit.

Resistivity and Resistance Relationship

• Resistivity Equation:

  • Resistance (R) can be defined as:

    • R = ρ * (l / A) (where ρ is resistivity, l is length, and A is cross-sectional area).

  • Effects of Area and Length on Resistance:

    • Increasing area decreases resistance (inversely related); for example, tripling area lowers resistance by a factor of nine (A increases by A^2).

    • Increasing length increases resistance (directly proportional).

Current and Resistance in Circuits

• Identifying Circuit Types:

  • If resistors are connected in series and parallel:

  • Example: Given resistors, determine total resistance and voltage across each.

  • Calculating Total Resistance:

    • For series: R_total = R1 + R2 + ...

    • For parallel: 1/R_total = 1/R1 + 1/R2 + ...

• Finding Current and Voltage:

  • Example: With a 6V battery and resistors, first find total resistance, then current using I = V / R.

  • Find voltage across each component in series using V = I * R.

Electrical Power in Circuits

• Power Equations:

  • P = V * I

  • P = I^2 * R

  • P = V^2 / R

  • These equations can be used interchangeably to find power consumption in different circuit configurations.

Efficiency and Thermodynamics

• Understanding Efficiency:

  • Efficiency: The proportion of useful energy output to total energy input, typically expressed as a percentage.

  • Common example: Internal combustion engines convert about 30% of energy input to useful work, rest lost as heat.

• Second Law of Thermodynamics:

  • States that in any energy transfer, the total entropy will increase; entropy is a measure of disorder.

Carnot Cycle and Reversibility

• Carnot Cycle:

  • An idealized thermodynamic cycle consisting of two isothermal (constant temperature) and two adiabatic (no heat exchange) processes. Each stage aims at maximum efficiency.

  • Concept of Reversibility: Carnot cycle is considered reversible as it theoretically cycles with no net increase in entropy.

Key Equations

• First Law of Thermodynamics (Conservation of Energy):

  • Total energy input (Q) into a gas = increase in internal energy (ΔU) + work done by the gas (W).

• Entropy Changes:

  • ΔS = Q / T (for processes at constant temperature).

  • Entropy measurement involves the Boltzmann constant and microstates: S = kB * ln(Ω).

  • Where Ω is the number of microstates related to a macrostate.