Electric Current, Resistance, Circuits, and Bioelectricity: Study Notes

Electric Current and Circuits

Introduction

Electricity is most useful when charges move in a controlled manner through conductors and circuit elements. This section introduces the foundational concepts of electric current, resistance, Ohm’s law, electric power, and the analysis of electric circuits, including their applications in biological systems.

Electric Current

Definition of Current

• Electric current (I) is the rate at which electric charge (ΔQ) flows through a cross-sectional area over time (Δt).

$I = \frac{\Delta Q}{\Delta t}$

• Unit: ampere (A), where 1 A = 1 C/s.

• By convention, current direction is the direction positive charge would move, even though electrons (negative charges) are the actual carriers in metals.

Charge Carriers in Metals

• In metals, electrons are the mobile charge carriers.

• Conventional current direction is opposite to electron flow.

Why Charges Move

• Charges move when an electric field exists inside the conductor, produced by a potential difference (voltage).

• The electric field points from higher to lower potential.

Resistance and Resistivity

Resistance

• Resistance (R) quantifies how strongly a material opposes charge flow.

$R = \frac{V}{I}$

• Unit: ohm (Ω).

• Higher resistance means less current for the same voltage.

Resistivity

• Resistivity (\rho) is a material property indicating how strongly a material resists current.

$R = \rho \frac{L}{A}$

• Longer wires have greater resistance; thicker wires have less.

• Good conductors (e.g., copper) have low resistivity; insulators have high resistivity.

Ohmic Materials

• Materials where current is proportional to voltage are called ohmic.

• For ohmic materials, the V-vs-I graph is linear, and resistance is constant.

Ohm’s Law

Statement of Ohm’s Law

• Relates voltage (V), current (I), and resistance (R):

$V = IR$

Equivalent forms: $I = \frac{V}{R}$ $R = \frac{V}{I}$

• Current increases with voltage and decreases with resistance.

Electric Power and Energy

Electric Power

• Electric power (P) is the rate of energy transfer or transformation in a circuit.

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

• Batteries supply power; resistors dissipate it as heat.

• Appliances convert electrical energy into other forms (e.g., heat, light, motion).

Batteries and Electromotive Force (emf)

Role of a Battery

• A battery maintains a potential difference (emf) between its terminals, pushing charge through the circuit.

• Emf is measured in volts (V).

Batteries in Series

• Voltages add when batteries are connected in series.

$V_{total} = V_1 + V_2 + \ldots$

• Example: Two 1.5 V cells in series provide 3.0 V.

Simple Circuit Concepts

• A basic circuit includes a source of emf, conducting path, and one or more loads (resistors, bulbs, etc.).

• Current flows only in a closed loop.

Series Circuits

Resistors in Series

• Same current passes through each resistor.

• Total resistance is the sum of individual resistances:

$R_{eq} = R_1 + R_2 + R_3 + \ldots$

• Voltages divide among resistors; equivalent resistance is greater than any individual resistance.

Example: Series Circuit

• Three resistors: $R_1 = 2\,\Omega$, $R_2 = 3\,\Omega$, $R_3 = 5\,\Omega$ in series with a 20 V battery.

$R_{eq} = 2 + 3 + 5 = 10\,\Omega$ $I = \frac{20}{10} = 2\,A$ $V_1 = (2)(2) = 4\,V$ $V_2 = (2)(3) = 6\,V$ $V_3 = (2)(5) = 10\,V$ $(Check: $4 + 6 + 10 = 20\,V$)

Parallel Circuits

Resistors in Parallel

• Each branch has the same voltage; current splits among branches.

• Equivalent resistance is found from:

$\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \ldots$

• Equivalent resistance is less than the smallest resistor.

Example: Parallel Circuit

• Two resistors: $R_1 = 6\,\Omega$, $R_2 = 3\,\Omega$ in parallel across 12 V.

$\frac{1}{R_{eq}} = \frac{1}{6} + \frac{1}{3} = \frac{1}{6} + \frac{2}{6} = \frac{3}{6} = \frac{1}{2}$ $R_{eq} = 2\,\Omega$ $I_{total} = \frac{12}{2} = 6\,A$ $I_1 = \frac{12}{6} = 2\,A$ $I_2 = \frac{12}{3} = 4\,A$ $(Check:$I_{total} = I_1 + I_2 = 6\,A$)

Comparing Series and Parallel Circuits

Property	Series	Parallel
Current	Same through all components	Divides among branches
Voltage	Divides among components	Same across each branch
Resistance	Increases as more resistors are added	Decreases as more branches are added
Failure	One failure opens the circuit	Other branches can remain on

Additional info: Holiday lights are often wired in series, while headlights are wired in parallel.

Conservation of Current

• Current is conserved at a junction: $I_{in} = I_{out}$.

• Charge is not used up; it flows continuously in a closed circuit.

Analyzing More Complex Circuits

Strategy

1. Identify series and parallel elements.

2. Replace with equivalent resistances.

3. Redraw the simplified circuit.

4. Repeat until only one equivalent resistance remains.

5. Use Ohm’s law to find total current.

6. Work backward to find currents and voltages in each component.

Example: Mixed Circuit

• $R_1 = 4\,\Omega$ in series with a parallel combination of $R_2 = 6\,\Omega$ and $R_3 = 3\,\Omega$, connected to 12 V.

$\frac{1}{R_{23}} = \frac{1}{6} + \frac{1}{3} = \frac{1}{2} \Rightarrow R_{23} = 2\,\Omega$ $R_{eq} = 4 + 2 = 6\,\Omega$ $I_{total} = \frac{12}{6} = 2\,A$ $V_1 = (2)(4) = 8\,V$ $V_{23} = 12 - 8 = 4\,V$ $I_2 = \frac{4}{6} = 0.67\,A$ $I_3 = \frac{4}{3} = 1.33\,A$ $(Check:$I_2 + I_3 = 2.00\,A$)

Capacitors in Circuits

Capacitance

• A capacitor stores charge and electric energy.

$C = \frac{Q}{V}$

Capacitors in Series

$\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + \ldots$

• Charge is the same on each capacitor; voltages add.

Capacitors in Parallel

$C_{eq} = C_1 + C_2 + \ldots$

• Voltage is the same across each; charges add.

Additional info: This is the opposite pattern from resistors.

RC Circuits

Time Constant

• The time constant $\tau$ determines how quickly a capacitor charges or discharges:

$\tau = RC$

• Larger $R$ or $C$ means slower response; smaller $R$ or $C$ means faster response.

Charging and Discharging

• Charging:

$q(t) = CV(1 - e^{-t/RC})$ $I(t) = \frac{V}{R} e^{-t/RC}$

• Discharging:

$q(t) = Q_0 e^{-t/RC}$ $I(t) = I_0 e^{-t/RC}$

• Capacitor behavior is exponential, not instantaneous.

Bioelectricity: The Electrical Nature of Nerve Cells

Membrane Potential

• A resting nerve cell has a potential difference (~70 mV) across its membrane (thickness ~7.0 nm).

• The electric field inside the membrane:

$E = \frac{\Delta V}{\Delta x}$ $E = \frac{70 \times 10^{-3}}{7.0 \times 10^{-9}} = 1.0 \times 10^7\,V/m$

• This is a very strong field due to the thin membrane.

Why the Membrane Acts Like a Capacitor

• Fluids inside and outside the cell are conductors; the membrane is an insulator.

• Charges accumulate on opposite sides, like a capacitor.

• The membrane also acts as a resistor due to ion channels, so it is modeled as an RC circuit.

Action Potential

• Membrane potential changes rapidly in response to a stimulus, producing an action potential.

• Resting state: stable potential difference.

• Depolarization: Sodium channels open, potential rises to +40 mV.

• Repolarization: Potassium channels open, potential drops to about −80 mV.

• Recovery: Ion motion restores resting potential.

• The action potential rise and fall is much faster than the RC time constant.

Propagation of Nerve Impulses

• Action potentials propagate as local changes in potential trigger neighboring ion channels.

• Myelin sheath insulates axons, increasing signal speed by reducing charge leakage and allowing impulses to jump between nodes of Ranvier.

Worked Examples

• Current from Charge Flow: 24 C in 8.0 s: $I = \frac{24}{8.0} = 3.0\,A$

• Resistance from Ohm’s Law: 2.0 A, 12 V: $R = \frac{12}{2.0} = 6.0\,\Omega$

• Power in a Resistor: 10 Ω, 3.0 A: $P = (3.0)^2(10) = 90\,W$

• Series Circuit: 4 Ω and 8 Ω in series, 24 V: $R_{eq} = 12\,\Omega$, $I = 2.0\,A$, voltage drops: 8 V and 16 V

• Parallel Circuit: 6 Ω and 12 Ω in parallel, 12 V: $R_{eq} = 4\,\Omega$, $I_{total} = 3.0\,A$, branch currents: 2.0 A and 1.0 A

• Electric Field in a Cell Membrane: 70 mV across 7.0 nm: $E = 1.0 \times 10^7\,V/m$

Common Student Difficulties

• Confusing current direction (conventional vs. electron flow).

• Mixing up resistor and capacitor rules (series/parallel addition).

• Thinking current is “used up” (it is conserved).

• Mixing up voltage and current in series/parallel circuits.

Summary of Key Equations

• Current: $I = \frac{\Delta Q}{\Delta t}$

• Ohm’s law: $V = IR$

• Resistance of a wire: $R = \rho \frac{L}{A}$

• Electric power: $P = IV$, $P = I^2 R$, $P = \frac{V^2}{R}$

• Resistors in series: $R_{eq} = R_1 + R_2 + \ldots$

• Resistors in parallel: $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \ldots$

• Capacitance: $C = \frac{Q}{V}$

• Capacitors in series: $\frac{1}{C_{eq}} = \frac{1}{C_1} + \ldots$

• Capacitors in parallel: $C_{eq} = C_1 + \ldots$

• RC time constant: $\tau = RC$

• Electric field from potential difference: $E = \frac{\Delta V}{\Delta x}$

Final Takeaways

Electric circuits are governed by a few powerful principles: charge flow, voltage difference, resistance, energy transfer, and equivalent circuit reduction. These concepts are foundational for physics, engineering, electronics, and physiology, especially in understanding biological systems such as nerve cells modeled as RC circuits.