Current Electricity: Exhaustive Study Notes

Introduction to Current Electricity

Charges in motion constitute an electric current. While previous studies have focused on charges at rest, current electricity examines the phenomena of flowing charges.

• Natural Occurrences: Lightning is a natural phenomenon where charges flow from clouds to the earth through the atmosphere. This flow is non-steady and can be disastrous.

• Steady Current: In everyday devices like torches or cell-driven clocks, charges flow in a steady manner, analogous to water flowing smoothly in a river.

Definition and Mathematical Representation of Electric Current

Electric current is defined by the flow of charge across a specific area held normal to the direction of flow.

• Net Charge Flow: If $q_+$ is the net positive charge flowing forward and $q_-$ is the net negative charge flowing forward in a time interval $t$, the total net charge is:     $q = q_+ - q_-$

• Steady Current Calculation: For a steady flow, current $I$ is proportional to time and is defined as:     $I = \frac{q}{t}$

• General/Instantaneous Current: For non-steady currents, the current at time $t$ is the limit of the ratio of charge $\triangle Q$ to the time interval $\triangle t$ as $\triangle t$ approaches zero:     I(t) \b\b≡ \b\b \b\blim_{\triangle t \to 0} \frac{\triangle Q}{\triangle t}

• SI Units: The unit of current is the ampere ($A$). It is defined through magnetic effects.

• Magnitude Examples:

  • Domestic appliances: Typical order of magnitude is amperes ($A$).

  • Lightning: Tens of thousands of amperes.

  • Human Nerves: Microamperes ($µA$).

Electric Currents in Conductors

Charges experience force in an electric field. If they are free to move, they create a current.

• Natural Free Carriers: Free charged particles exist in the ionosphere (upper atmosphere).

• Atomic Structure: In bulk matter, electrons and nuclei are bound. A gram of water contains approximately $10^{22}$ molecules.

• Classification:

  • Conductors: Materials where some electrons are practically free to move within the bulk material (e.g., metals).

  • Insulators: Materials where electrons remain bound even under an applied electric field.

• Solid Conductors: Current is carried by negatively charged electrons against a background of fixed positive ions.

• Electrolytic Solutions: Both positive and negative charges can move to carry current.

• Thermal Motion: In the absence of an electric field, electrons move randomly due to thermal energy, colliding with fixed ions. Their average velocity is zero, resulting in no net current.

• Mechanism for Steady Current: Applying an electric field (e.g., via a battery or cell) accelerates electrons. Transient currents occur when charges move to neutralize a field (like on dielectric discs); steady currents require a mechanism to replenish charges continuously.

Ohm’s Law and Resistance

Discovered by G.S. Ohm in 1828, the law relates current and potential difference.

• The Law: The potential difference $V$ across the ends of a conductor is proportional to the current $I$ flowing through it:     V ∝ I \b\b \b\b \to \b\b \b\b V = RI

• Resistance ($R$): The constant of proportionality, measured in ohms ($Ω$).

• Standard Analogy: Ohm developed his law using an analogy with heat conduction, where electric field corresponds to temperature gradient and current to heat flow.

Dependence of Resistance on Dimensions

• Length ($l$): For two identical slabs placed side by side (length becomes $2l$), the potential difference doubles for the same current. Thus, $R ∝ l$.

• Area ($A$): If a slab is cut lengthwise (area becomes $A/2$), the current through each half is $I/2$ for the same voltage. Thus, resistance doubles when area is halved: $R ∝ \frac{1}{A}$.

• Combined Relation:     $R = \rho \frac{l}{A}$     where $\rho$ is the resistivity, a material-specific constant independent of dimensions.

Current Density and Conductivity

• Current Density ($j$): Current per unit area normal to the flow ($j = \frac{I}{A}$). SI unit: $A/m^2$.

• Vector Form of Ohm's Law: If $E$ is the uniform electric field ($V = El$), then:     E = j \rho \b\b \b\b \text{or} \b\b \b\b j = \b\b \b\b \text{σ} E     where conductivity $\text{σ} = \frac{1}{\rho}$.

Drift of Electrons and Origin of Resistivity

Electrons in a conductor undergo collisions with heavy fixed ions.

• Acceleration: Under an electric field $E$, an electron of charge $-e$ and mass $m$ accelerates:     $a = - \frac{eE}{m}$

• Relaxation Time ($τ$): The average time elapsed between successive collisions.

• Drift Velocity ($v_d$): The average velocity acquired by electrons due to the electric field. It is independent of time despite constant acceleration because of collisions:     $v_d = - \frac{eEτ}{m}$

• Charge Transport: In time $\triangle t$, the charge crossing area $A$ is:     $\triangle Q = neA |v_d| \triangle t$     where $n$ is the number of free electrons per unit volume.

• Conductivity Expression: Identifying $j = \text{σ} E$ with $I = neA|v_d|$ yields:     $\text{σ} = \frac{ne^2τ}{m}$

• Resistivity Expression:     $\rho = \frac{m}{ne^2τ}$

Practical Comparisons (Example 3.1 & 3.2)

• Drift Speed in Copper: Typically \b≈ 1.1 \b\b \times 10^{-3} \b\b m/s (or 1.1 \b\b mm/s) for a current of 1.5 \b\b A in a wire with area 1.0 \b\b \times 10^{-7} \b\b m^2.

• Number Density in Copper: n \b≈ 8.5 \b\b \times 10^{28} \b\b m^{-3}.

• Thermal Speed: Approximately 2 \b\b \times 10^2 \b\b m/s. Drift speed is $10^{-5}$ times smaller.

• Propagation Speed: The electric field travels at the speed of electromagnetic waves (3.0 \b\b \times 10^8 \b\b m/s), which is why current is established instantly.

• Electron Paths: Straight lines between collisions without a field; curved paths with a field.

Mobility

Mobility ($μ$) measures how quickly a charge carrier moves through a metal or semiconductor when pulled by an electric field.

• Definition: Magnitude of drift velocity per unit electric field:     $μ = \frac{|v_d|}{E} = \frac{eτ}{m}$

• Units: SI unit is $m^2/Vs$.

• Practical Context: In metals, carriers are electrons; in ionized gases, they are electrons and positive ions; in electrolytes, they are positive and negative ions.

Limitations of Ohm’s Law

Ohm's law is not a universal law and fails for certain materials and devices:

1. Non-linear regions: In many good conductors, $V$ and $I$ cease to be proportional at high currents (thermal effects).

2. Sign Dependence: In devices like diodes, reversing the polarity of $V$ does not produce an equal magnitude of current $I$.

3. Non-unique V-I Relationship: In materials like GaAs, a single current value can correspond to multiple potential difference values.

Resistivity and Temperature Dependence

Materials are classified by their resistivities:

• Metals: High conductivity; resistivity 10^{-8} \b\b Ωm to 10^{-6} \b\b Ωm.

• Insulators: Resistivity $10^{18}$ times higher than metals.

• Semiconductors: Decrease in resistivity with increasing temperature.

Temperature Coefficient of Resistivity ($α$)

For metallic conductors over limited temperature ranges: $\rho_T = \rho_0 [1 + α(T - T_0)]$

• Metals: $α$ is positive (resistivity increases with temperature). Curves deviate from linearity at very low temperatures.

• Alloys (Nichrome, Manganin, Constantan): Very weak temperature dependence. Used in wire-bound standard resistors.

• Semiconductors/Insulators: $α$ is negative (resistivity decreases with temperature). This is because the charge carrier density $n$ increases exponentially with temperature, overcoming the decrease in relaxation time $τ$.

Electrical Energy and Power

When a charge $\triangle Q = I \triangle t$ moves across a potential difference $V$, its potential energy changes by: $\triangle U_{pot} = -I V \triangle t$

• Energy Dissipation: In a conductor, this energy is not converted solely to kinetic energy (due to collisions) but is dissipated as heat.

• Power ($P$): Energy dissipated per unit time:     $P = VI = I^2 R = \frac{V^2}{R}$

• Power Transmission: To minimize waste power ($P_c$) in cables with resistance $R_c$ over long distances, electricity is transmitted at high voltages:     $P_c = \frac{P^2 R_c}{V^2}$     High voltage reduces current, thus reducing "ohmic loss."

Cells, EMF, and Internal Resistance

An electrolytic cell maintains a steady current by moving charges from lower to higher potential energy using chemical energy.

• EMF ($ε$): The potential difference between the positive (P) and negative (N) electrodes when no current is flowing (open circuit).     ε = V_+ + V_- > 0

• Internal Resistance ($r$): The resistance offered by the electrolyte inside the cell.

• Terminal Voltage ($V$): When current $I$ flows:     $V = ε - Ir$

• Circuit Current: For a cell connected to external resistance $R$:     $I = \frac{ε}{R + r}$

• Maximum Current: Occurs when $R = 0$, so $I_{max} = \frac{ε}{r}$.

Combinations of Cells

Cells in Series

For $n$ cells connected such that the negative terminal of one joins the positive terminal of the next:

• Equivalent EMF: ε_{eq} = ε_1 + ε_2 + \b… + ε_n

• Equivalent Internal Resistance: r_{eq} = r_1 + r_2 + \b… + r_n

• Opposing Cells: If a cell is connected with reversed polarity, its EMF enters the sum with a negative sign.

Cells in Parallel

For cells connected with positive terminals joined and negative terminals joined:

• General Formula:     \frac{1}{r_{eq}} = \frac{1}{r_1} + \frac{1}{r_2} + \b…     \frac{ε_{eq}}{r_{eq}} = \frac{ε_1}{r_1} + \frac{ε_2}{r_2} + \b…

• For Two Cells:     $ε_{eq} = \frac{ε_1 r_2 + ε_2 r_1}{r_1 + r_2}$     $r_{eq} = \frac{r_1 r_2}{r_1 + r_2}$

Kirchhoff’s Rules

Essential for analyzing complex circuits where series/parallel simplifications fail.

1. Junction Rule: At any junction, the sum of currents entering equals the sum of currents leaving. This is based on the conservation of charge.

2. Loop Rule: The algebraic sum of changes in potential around any closed loop is zero. This is based on the conservation of energy.

Wheatstone Bridge

An arrangement of four resistors ($R_1, R_2, R_3, R_4$) used to determine an unknown resistance.

• Structure: Diagonal AC contains a battery (battery arm); diagonal BD contains a galvanometer (galvanometer arm).

• Balance Condition: The bridge is balanced when current through the galvanometer $I_g = 0$.

• Condition derivation:

  1. Junction rule implies $I_1 = I_3$ and $I_2 = I_4$.

  2. Loop rule for the two halves of the bridge results in:         $\frac{R_2}{R_1} = \frac{R_4}{R_3}$

• Practical Use: If $R_4$ is unknown, it can be calculated as $R_4 = R_3 \frac{R_2}{R_1}$ by varying known resistors until null deflection is achieved in the galvanometer.

Physical Quantities and Units Summary

• Electric Current ($I$): $[A]$, Unit: Ampere ($A$).

• Charge ($Q$): $[TA]$, Unit: Coulomb ($C$).

• Voltage/Potential Difference ($V$): $[ML^2T^{-3}A^{-1}]$, Unit: Volt ($V$).

• Electromotive Force ($ε$): $[ML^2T^{-3}A^{-1}]$, Unit: Volt ($V$).

• Resistance ($R$): $[ML^2T^{-3}A^{-2}]$, Unit: Ohm ($Ω$).

• Resistivity ($ρ$): $[ML^3T^{-3}A^{-2}]$, Unit: $Ωm$.

• Electrical Conductivity ($\text{σ}$): $[M^{-1}L^{-3}T^3A^2]$, Unit: Siemens ($S$).

• Electric Field ($E$): $[MLT^{-3}A^{-1}]$, Unit: $V/m$.

• Drift Speed ($v_d$): $[LT^{-1}]$, Unit: $m/s$.

• Relaxation Time ($τ$): $[T]$, Unit: Second ($s$).

• Current Density ($j$): $[L^{-2}A]$, Unit: $A/m^2$.

• Mobility ($μ$): $[M^{-1}T^2A]$, Unit: $m^2 V^{-1} s^{-1}$.

Questions & Discussion

Q: How is current established instantly when drift speed is so slow? An electric field is established throughout the circuit almost at the speed of light, causing local electron drift everywhere simultaneously. Current does not wait for a single electron to travel from one end to the other.

Q: Why do electrons acquire a steady drift speed instead of accelerating indefinitely? Each electron accelerates under the electric field but loses its acquired velocity during collisions with positive ions. This repeated process results in a constant average drift speed.

Q: Can a large current be obtained with small drift speeds and charge? Yes, because the number density of free electrons in metals is enormous, roughly 10^{29} \b\b m^{-3}.

Q: Are electron paths straight lines? In the absence of an electric field, paths between collisions are straight lines. In the presence of a field, paths are generally curved.

Q: Is current a vector? Current is a scalar. While represented with arrows in diagrams, it does not obey laws of vector addition. It is defined as the scalar product of current density and area vectors (I = j \b\b \b\b ḅ \b\b \b\b \triangle S).