Electrical Properties: Current, Resistance, Ohm's Law, and Capacitors

Electrical Properties: Current, Resistance, Ohm's Law, and Capacitors

Learning Objectives

• Understand the nature of electric charge and interactions between charges.

• Describe electric current as the rate of flow of charge and understand its units.

• Define voltage (potential difference) and its significance.

• Apply the concept of resistance and use Ohm's Law in calculations.

• Recognize conductance as the reciprocal of resistance.

• Analyze combinations of resistors and conductors in series and parallel.

• Understand the function and properties of capacitors, including charging and discharging behavior.

• Relate physical concepts to biological systems, such as the cell membrane as a capacitor.

Electrical Symbols

• Cell: Provides a source of electrical energy.

• Battery: Combination of cells for higher voltage.

• Resistor: Limits current flow.

• Variable Resistor: Resistance can be adjusted.

• Capacitor: Stores electrical charge.

• Switch: Opens or closes a circuit.

Electric Current

Definition and Units

• Electric current is the movement of charge through a conductor.

• Defined as the rate at which charge passes a point in a closed circuit.

• Measured in amperes (A): 1 ampere = 1 coulomb/second.

• Symbol: I (sometimes i in physiology).

Equation:

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

• Where Q is charge (in coulombs), t is time (in seconds).

Resistance and Conductance

Resistance

• Resistance (R) is the opposition to the flow of charge through a conductor.

• Measured in ohms (\Omega).

• Symbol in equations: R.

• Materials that prevent current flow are called electrical insulators.

Conductance

• Conductance (G) is the ease with which charge flows through a conductor.

• It is the reciprocal of resistance: $G = \frac{1}{R}$

• Measured in siemens (S).

• Symbol in equations: G (sometimes g).

Ohm's Law

Relationship Between Voltage, Current, and Resistance

• Ohm's Law quantifies the relationship between potential difference (V), current (I), and resistance (R):

$V = IR$

• Rearrangements:

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

• Using conductance:

$V = IG$ $G = \frac{I}{V}$

Example Calculations

• Example 1: A current of 0.5 A flows through a resistor of 10 \Omega. What is the voltage?

$V = IR = 0.5 \times 10 = 5\ \text{V}$

• Example 2: A potential difference of 1.5 V is applied across a 150 \Omega resistor. What is the current?

$I = \frac{V}{R} = \frac{1.5}{150} = 0.01\ \text{A} = 10\ \text{mA}$

• Example 3: A potential difference of 3 V is applied across a resistor. The current is 24 mA (0.024 A). What is the resistance?

$R = \frac{V}{I} = \frac{3}{0.024} = 125\ \Omega$

Resistors in Series and Parallel

Resistors in Series

• For resistors in series, the total resistance is the sum of individual resistances:

$R_T = R_1 + R_2 + R_3 + \ldots + R_N$

• The same current flows through each resistor.

• The total voltage is divided among the resistors.

Example Calculation

• To generate a current of 20 mA (0.02 A) across resistors $R_1 = 200\ \Omega$ and $R_2 = 600\ \Omega$ in series, what voltage is needed?

$R_T = 200 + 600 = 800\ \Omega$ $V = IR = 0.02 \times 800 = 16\ \text{V}$

Resistors in Parallel

• For resistors in parallel, the reciprocal of the total resistance is the sum of the reciprocals of individual resistances:

$\frac{1}{R_T} = \frac{1}{R_1} + \frac{1}{R_2} + \ldots + \frac{1}{R_N}$

• Alternatively, using conductance:

$G_T = G_1 + G_2 + \ldots + G_N$

• The total current is the sum of the currents through each resistor.

Example Calculation

• For $R_1 = 100\ \Omega$, $R_2 = 200\ \Omega$ in parallel:

$\frac{1}{R_T} = \frac{1}{100} + \frac{1}{200} = \frac{3}{200}$ $R_T = \frac{200}{3} = 66.66\ \Omega$

• With a 6 V battery, the current is:

$I = \frac{V}{R_T} = \frac{6}{66.66} = 0.09\ \text{A} = 90\ \text{mA}$

Capacitors

Definition and Function

• A capacitor is an electric device that stores charge.

• The simplest form is a parallel plate capacitor with two conducting plates separated by an insulating material (dielectric).

Charging and Discharging

• When connected to a battery, electrons flow from the negative terminal to one plate, creating a charge separation.

• Charge accumulation generates a potential difference across the plates.

• Charging current decreases as the capacitor approaches the battery voltage.

• When disconnected, the capacitor can discharge through a closed circuit.

Charging/Discharging Behavior

• Charging and discharging follow an exponential time course.

• Potential difference and current change over time as the capacitor charges or discharges.

Properties of a Parallel Plate Capacitor

• The capacitance (C) is the amount of charge stored per unit voltage, measured in Farads (F).

• Capacitance depends on:

  • Surface area of the plates (A): increases with area.

  • Distance between plates (d): decreases with increasing distance.

  • Properties of the dielectric (\( \varepsilon_r \)): relative permittivity.

Equation:

$C = \varepsilon_0 \varepsilon_r \frac{A}{d}$

• Where $\varepsilon_0$ is the electric constant ($8.854 \times 10^{-12}\ \text{F m}^{-1}$), $\varepsilon_r$ is the relative permittivity (1 for air), $A$ is area, $d$ is separation.

Applications and Biological Context

• The cell (plasma) membrane acts as a capacitor, maintaining a charge gradient essential for nerve function, ATP synthesis, and cell signaling.

• Separated charges across the membrane create a potential difference (resting membrane potential).

Example: Application of Ohm's Law in Biology

• To determine the input resistance of a neuron, a current is injected and the change in membrane potential is measured.

• Calculation:

$R = \frac{V}{I} = \frac{15 \times 10^{-3}}{0.5 \times 10^{-9}} = 30 \times 10^6\ \Omega = 30\ \text{M}\Omega$

Important Terms and Units

Abbreviation	Physical Property	Unit	Unit Symbol
Q, q	charge	coulomb	C
I, i	current	ampere	A
V	electrical potential (potential difference)	volt	V
R, r	resistance	ohm	Ω
G	conductance	siemens	S
ρ (rho)	resistivity	ohm·metre	Ω·m
C, c	capacitance	farad	F

Additional info: The notes include biological applications to illustrate the relevance of electrical properties in physiology, particularly in nerve and cell membrane function.