Current Electricity Practice Flashcards

Introduction to Current Electricity

• Conceptual Overview: In previous studies, both free and bound charges were considered at rest. Charges in motion constitute an electric current.

• Natural vs. Steady Currents:     * Natural Phenomena: Lightning is a non-steady flow of charges from clouds to the earth through the atmosphere, often with disastrous effects.     * Everyday Devices: Devices such as torches and cell-driven clocks exhibit steady currents, comparable to the smooth flow of water in a river.

Electric Current Definition and Units

• Mathematical Definition: Consider a small area normal to the direction of charge flow. If $q_+$ is the net positive charge flowing forward and $q_-$ is the net negative charge flowing forward in time $t$, the net charge $q$ is:     $q = q_+ - q_-$

• Steady Current: The current $I$ across the area is the quotient:     $I = \frac{q}{t}$

• General/Varying Current: For a net charge $\Delta Q$ flowing across a cross-section in time interval $\Delta t$ (between times $t$ and $t + \Delta t$), the instantaneous current is:     $I(t) = \lim_{\Delta t \rightarrow 0} \frac{\Delta Q}{\Delta t}$

• SI Unit: The unit of current is the ampere (A), defined through magnetic effects of currents.

• Orders of Magnitude:     * Domestic Appliances: ~$1\,A$     * Average Lightning: Tens of thousands of amperes ($10^4\,A$).     * Human Nerves: Microamperes ($\mu A$).

Electric Currents in Conductors

• Mechanism: An electric field applies a force to charges; if free to move, they contribute to a current.

• Free Particles: Exist in the ionosphere. However, in bulk matter (e.g., a gram of water containing approximately $10^{22}$ molecules), electrons and nuclei are usually bound.

• Metals (Conductors): Electrons are practically free to move within the bulk material. In 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.

• Case 1: No Electric Field: Electrons undergo thermal motion and collide with fixed ions. Their speed remains constant after collision, but direction is random. The average velocity is zero:     $\frac{1}{N} \sum_{i=1}^{N} \mathbf{v}_i = 0$

• Case 2: Application of Electric Field: If a cylinder of radius $R$ has dielectric discs with charges $+Q$ and $-Q$ attached to its ends, an electric field is created. Electrons accelerate toward $+Q$, creating a transient current until charges are neutralized.

• Steady Field: Maintained by cells or batteries which replenish charges at the ends of the conductor.

Ohm’s Law and Resistance

• Law (G.S. Ohm, 1828): For a conductor with current $I$ and potential difference $V$ across its ends:     $V \propto I \implies V = RI$

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

• Factors Governing Resistance:     * Length ($l$): Resistance is directly proportional to length: $R \propto l$.     * Cross-sectional Area ($A$): Resistance is inversely proportional to area: $R \propto \frac{1}{A}$.     * Formula: $R = \rho \frac{l}{A}$

• Resistivity ($\rho$): A material constant independent of dimensions. SI unit: $\Omega\,m$.

Current Density, Conductivity, and Vector Form of Ohm's Law

• Current Density ($j$): Current per unit area normal to flow:     $j = \frac{I}{A}$

• Potential Difference and Field ($E$): For a uniform field across length $l$:     $V = El$

• Ohm's Law in terms of Field: From $V = I \rho \frac{l}{A}$, we get $El = (j A) \rho \frac{l}{A} \implies E = j \rho$.

• Vector Form: Since current density is along the direction of $E$:     $\mathbf{E} = \rho \mathbf{j} \implies \mathbf{j} = \sigma \mathbf{E}$

• Electrical Conductivity ($\sigma$): Defined as the reciprocal of resistivity:     $\sigma = \frac{1}{\rho}$

Drift of Electrons and the Origin of Resistivity

• Microscopic Mechanism: Electrons accelerate under an electric field between collisions with heavy ions.

• Acceleration: $\mathbf{a} = \frac{-e\mathbf{E}}{m}$

• Relaxation Time (\tau): The average time between successive collisions.

• Velocity after Collision: If $\mathbf{v}_i$ is the velocity immediately after the last collision at time $t_i$ ago, the velocity $\mathbf{V}_i$ at time $t$ is:     $\mathbf{V}_i = \mathbf{v}_i - \frac{e\mathbf{E}}{m} t_i$

• Drift Velocity ($v_d$): The average velocity of all electrons. Since the average of initial thermal velocities is zero, and the average of $t_i$ is $\tau$:     $\mathbf{v}_d = -\frac{e\mathbf{E}\tau}{m}$

• Link to Current: In time $\Delta t$, all electrons within distance $|v_d|\Delta t$ cross area $A$. If $n$ is the free electron number density:     $I \Delta t = n e A |v_d| \Delta t \implies I = n e A v_d$

• Identifying Conductivity:     $j = n e v_d = \frac{n e^2 \tau}{m} E \implies \sigma = \frac{n e^2 \tau}{m}$     $\rho = \frac{m}{n e^2 \tau}$

Examples and Comparisons

• Example 3.1: Copper Wire:     * Data: $A = 1.0 \times 10^{-7}\,m^2$, $I = 1.5\,A$, Density = $9.0 \times 10^3\,kg/m^3$, Atomic mass = $63.5\,u$.     * Free electron density ($n$): $8.5 \times 10^{28}\,m^{-3}$.     * Calculated Drift Speed ($v_d$): $1.1\,mm/s$.

• Comparison of Speeds:     * Drift Speed: $\sim 10^{-3}\,m/s$.     * Thermal Speed: $\sim 2 \times 10^2\,m/s$ (at $300\,K$).     * Speed of EM waves: $3.0 \times 10^8\,m/s$ (speed at which current is established).

Mobility

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

• Equation: $\mu = \frac{e \tau}{m}$

• SI Unit: $m^2\,V^{-1}\,s^{-1}$.

Limitations of Ohm’s Law

• Non-proportionality: In many conductors, $V$ stops being proportional to $I$ at high currents (heating effects).

• Polarity Dependence: The relation between $V$ and $I$ depends on the sign of $V$ (e.g., semiconductor diodes).

• Non-uniqueness: Materials like Gallium Arsenide (GaAs) exhibit regions where the same current $I$ can correspond to different voltages $V$

Resistivity and Temperature Dependence

• Classification:     * Metals: Low resistivity ($10^{-8}$ to $10^{-6}\,\Omega m$).     * Insulators: High resistivity ($10^{14}$ to $10^{16}\,\Omega m$ or more).     * Semiconductors: Intermediate range; resistivity decreases with temperature rise.

• Metallic Temperature Coefficient (\alpha):     $\rho_T = \rho_0 [1 + \alpha(T - T_0)]$     * $\alpha$ is the temperature coefficient of resistivity ($K^{-1}$ or $^{\circ}C^{-1}$).     * For metals, $\alpha$ is positive. Alloys like Nichrome, Manganin, and Constantan have extremely small $\alpha$, making them ideal for standard resistors.

• Explanation: In metals, increasing $T$ increases thermal speed, leading to more frequent collisions and smaller $\tau$, thus increasing $\rho$. In semiconductors/insulators, the number density $n$ increases exponentially with temperature, which outweighs the decrease in $\tau$, causing $\rho$ to decrease.

Electrical Energy and Power

• Potential Energy Change: A charge $\Delta Q = I \Delta t$ moving through potential difference $V$ loses potential energy:     $\Delta U_{pot} = -I V \Delta t$

• Heat Dissipation (Ohmic Loss): This energy is converted into thermal vibrations of ions:     $\Delta W = I V \Delta t$

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

• Power Transmission: To minimize power loss ($P_c = I^2 R_c$) in transmission cables of resistance $R_c$ over long distances:     $P_c = \frac{P^2 R_c}{V^2}$     High voltage ($V$) significantly reduces power waste ($P_c$).

Cells, EMF, and Internal Resistance

• Electromotive Force (\varepsilon): The potential difference between positive ($P$) and negative ($N$) electrodes of a cell in an open circuit (no current).     $\varepsilon = V_+ + V_-$

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

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

• Current in External Resistor ($R$):     $I = \frac{\varepsilon}{R + r}$

• Maximum Current: $I_{max} = \frac{\varepsilon}{r}$ (when $R = 0$).

Combination of Cells

• Series Combination:     * $\varepsilon_{eq} = \sum \varepsilon_i$     * $r_{eq} = \sum r_i$     * If a cell is connected with reversed polarity, its emf enters with a negative sign.

• Parallel Combination (Two Cells):     * $\frac{1}{r_{eq}} = \frac{1}{r_1} + \frac{1}{r_2} \implies r_{eq} = \frac{r_1 r_2}{r_1 + r_2}$     * $\frac{\varepsilon_{eq}}{r_{eq}} = \frac{\varepsilon_1}{r_1} + \frac{\varepsilon_2}{r_2} \implies \varepsilon_{eq} = \frac{\varepsilon_1 r_2 + \varepsilon_2 r_1}{r_1 + r_2}$

Kirchhoff’s Rules

• Junction Rule: At any junction, the sum of currents entering equals the sum of currents leaving ($\sum I = 0$). Based on conservation of charge.

• Loop Rule: The algebraic sum of changes in potential around any closed loop is zero ($\sum \Delta V = 0$). Based on conservation of energy.

• Example 3.5: Cubical Network: 12 resistors of resistance $R$ form a cube. For diagonally opposite corners, equivalent resistance is:     $R_{eq} = \frac{5}{6}R$

Wheatstone Bridge

• Structure: Four resistors ($R_1, R_2, R_3, R_4$) in a bridge with a galvanometer in the bridge arm ($BD$) and a battery in the battery arm ($AC$).

• Balanced Condition: When the galvanometer current $I_g = 0$:     $\frac{R_1}{R_2} = \frac{R_3}{R_4} \implies \frac{R_2}{R_1} = \frac{R_4}{R_3}$

• Application: Determination of unknown resistance $R_4 = R_3 \frac{R_2}{R_1}$.

Questions & Discussion

• Question: How is current established instantly if drift speed is slow?

• Answer: An electric field is established throughout the circuit at nearly the speed of light, causing local electrons everywhere to drift almost immediately.

• Question: Why steady drift speed instead of acceleration?

• Answer: Frequent collisions with positive ions neutralize the acceleration, resulting in a steady average velocity.

• Question: Why large currents if drift speed is small?

• Answer: Because the number density of free electrons is enormous ($\sim 10^{29}\,m^{-3}$).

• Question: Are paths straight between collisions?

• Answer: Straight lines in the absence of an electric field; curved in the presence of an electric field.