Study Notes on Current Electricity

Introduction to Electric Current

• Conceptual Overview: In previous studies (Chapter 1), charges were considered at rest. This chapter focuses on charges in motion, which constitute an electric current.

• Natural Occurrence: Current occurs in nature, such as in lightning, where charges flow from clouds to the earth through the atmosphere. Lightning is an example of non-steady current.

• Steady Current Examples: Common devices utilize steady current, where charges flow smoothly like water in a river. Examples include a torch and a cell-driven clock.

• Objective: To study the basic laws governing steady electric currents.

Electric Current Definition and SI Units

• General Definition: Imagine a small area held normal to the direction of flow. Let $q_+$ be the net positive charge flowing forward minus backward in time $t$, and $q_-$ be the net negative charge flowing across the same area in the forward direction. The net charge flowing across the area in time $t$ is $q = q_+ - q_-$.

• Steady Current Formula: For current that does not vary with time:     $I = \frac{q}{t}$

• Varying Current Definition: If the flow of charge varies with time, we define the current at time $t$ as the limit of the ratio of charge $\Delta Q$ to time interval $\Delta t$ as $\Delta t$ tends to zero:     $I(t) = \lim_{\Delta t \to 0} \frac{\Delta Q}{\Delta t} = \frac{dQ}{dt}$

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

• Orders of Magnitude:

  • Domestic Appliances: Typically of the order of $1\,A$.

  • Average Lightning: Involves currents of tens of thousands of amperes ($10^4\,A$).

  • Human Nerves: Currents are in the range of microamperes ($10^{-6}\,A$).

Electric Currents in Conductors

• Mechanism of Flow: Charges experience force in an electric field. If free to move, they create a current. Free particles exist in the ionosphere; however, in bulk matter, electrons and nuclei are usually bound in atoms/molecules.

• Bulk Matter Concentration: A gram of water contains approximately $10^{22}$ molecules.

• Conductors vs. Insulators:

  • Conductors: Materials (notably metals) where some electrons are practically free to move within the bulk. Atoms are tightly bound, and current is carried by negatively charged electrons.

  • Insulators: Materials where electrons remain bound and do not accelerate under an applied electric field.

  • Electrolytic Solutions: Conductors where both positive and negative charges can move.

• Case 1: No Electric Field: Electrons undergo thermal motion and collide with fixed ions. After collision, they emerge with the same speed but in random directions. The average number of electrons traveling in any direction equals those in the opposite direction; thus, net current is zero.

• Case 2: Applied Electric Field: Consider a cylinder of radius $R$. If two circular dielectric discs with charges $+Q$ and $-Q$ are attached to the ends, an electric field is created. Electrons accelerate toward $+Q$ to neutralize the charges. To maintain a steady current, charges must be continuously replenished by mechanisms like cells or batteries.

Ohm’s Law and Electrical Resistance

• Origin: Discovered by G.S. Ohm in 1828. It relates current ($I$) and potential difference ($V$) across a conductor.

• The Law: For many materials, the potential difference is proportional to the current:     $V \propto I$     $V = R I$

• Resistance ($R$): The constant of proportionality. Its SI unit is the ohm ($\Omega$).

• Dependence on Dimensions:

  • Length ($l$): Resistance is directly proportional to length. Doubling the length (placing two identical slabs side-by-side) doubles the potential difference for the same current, so $R \propto l$.

  • Area ($A$): Resistance is inversely proportional to cross-sectional area. Halving the area (splitting a slab lengthwise) doubles the resistance for the same voltage, so $R \propto \frac{1}{A}$.

• Resistivity ($\rho$): Combining dependencies:     $R = \rho \frac{l}{A}$     where $\rho$ is the resistivity, a property of the material dependent on temperature but independent of dimensions.

Microscopic Form of Ohm’s Law and Current Density

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

• Potential Difference and Field: For a uniform electric field ($E$) in a conductor of length $l$:     $V = E l$

• Relation between E and j:     $E l = (j A) \times (\rho \frac{l}{A})$     $E = j \rho$

• Vector Notation: Current density $\mathbf{j}$ is a vector directed along $\mathbf{E}$. Thus:     $\mathbf{E} = \mathbf{j} \rho$     $\mathbf{j} = \sigma \mathbf{E}$     where $\sigma = \frac{1}{\rho}$ is the electrical conductivity.

Drift of Electrons and the Origin of Resistivity

• Electron Dynamics: Electrons accelerate in an electric field with acceleration:     $\mathbf{a} = \frac{-e\mathbf{E}}{m}$     where $-e$ is the charge and $m$ is the mass of an electron.

• Relaxation Time ($\tau$): The average time interval between successive collisions of an electron with fixed ions.

• Drift Velocity ($v_d$): The average velocity acquired by electrons due to the electric field despite random collisions:     $\mathbf{v}_d = \frac{-e\mathbf{E}\tau}{m}$

• Current and Drift Velocity Relation: The charge transported across area $A$ in time $\Delta t$ is $\Delta Q = n e A |v_d| \Delta t$, where $n$ is the number of free electrons per unit volume. The magnitude of current is:     $I = n e A |v_d|$

• Conductivity Formula: Substituting $|v_d| = \frac{eE\tau}{m}$ into $I/A = j$:     $j = (\frac{n e^2 \tau}{m}) E$     Thus, $\sigma = \frac{n e^2 \tau}{m}$ and $\rho = \frac{m}{n e^2 \tau}$.

Mobility

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

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

• Mobility Relation:     $\mu = \frac{e\tau}{m}$

• Units: SI unit is $m^2/Vs$. Mobility is always positive.

Example 3.1: Drift Speed in Copper

• Data: $A = 1.0 \times 10^{-7}\,m^2$, $I = 1.5\,A$, Density of Copper = $9.0 \times 10^3\,kg/m^3$, Atomic Mass = $63.5\,u$.

• Number Density ($n$): Calculated as $n = 8.5 \times 10^{28}\,m^{-3}$.

• Calculation:     $v_d = \frac{I}{neA} = 1.1 \times 10^{-3}\,m/s = 1.1\,mm/s$

• Comparisons:

  • Thermal speed of Cu atoms: Roughly $2 \times 10^2\,m/s$ at $300\,K$. Drift speed is $10^{-5}$ times smaller.

  • Electric Field propagation: Speed of electromagnetic waves ($3.0 \times 10^8\,m/s$). Drift speed is $10^{-11}$ times smaller.

Limitations of Ohm’s Law

• Deviations: Ohm's law is not a fundamental law and fails in several cases:

  • Non-linear V-I relationship: Voltage is not proportional to current at high currents or in specific conductors.

  • Directional dependence: The current magnitude depends on the sign (direction) of voltage, e.g., in a junction diode.

  • Non-unique V for I: Multiple voltages can produce the same current, e.g., in Gallium Arsenide (GaAs).

Temperature Dependence of Resistivity

• Relationship for Metals: Over limited temperature ranges:     $\rho_T = \rho_0 [1 + \alpha(T - T_0)]$     where $\alpha$ is the temperature coefficient of resistivity.

• Coefficient $\alpha$:

  • Positive for metals.

  • Negative for semiconductors and insulators.

• Behavior by Material:

  • Metals: $\rho$ increases with $T$ because relaxation time $\tau$ decreases as electrons collide more frequently with vibrating ions.

  • Alloys (Nichrome, Manganin, Constantan): Exhibit very weak temperature dependence; used for standard resistors.

  • Semiconductors/Insulators: $\rho$ decreases as $T$ increases because the number density of carriers ($n$) increases significantly with temperature, overcoming the decrease in $\tau$.

Electrical Energy and Power

• Potential Energy Change: As charge $\Delta Q = I \Delta t$ moves from point A to B through potential difference V = V(A) - V(B) > 0:     $\Delta U_{pot} = -IV \Delta t$

• Energy Dissipation: In conductors, kinetic energy gained between collisions is transferred to atoms as heat. Energy dissipated in time $\Delta t$ is:     $W = IV \Delta t$

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

• Power Transmission: To minimize energy loss ($P_c = I^2 R_c$) in long cables, power is transmitted at high voltage ($V$) and low current ($I$) because $P_c \propto \frac{1}{V^2}$. Transformers then step-down voltage for safe use.

Cells, EMF, and Internal Resistance

• The Electrolytic Cell: Maintains steady current by moving charges from lower to higher potential using chemical energy.

• Electromotive Force (EMF, $\varepsilon$): The potential difference between terminal electrodes when no current is flowing (open circuit):     \varepsilon = V_+ + V_- > 0

• Internal Resistance ($r$): The inherent resistance of the electrolyte and electrodes within the cell.

• Terminal Voltage ($V$): When current $I$ flows, the potential difference between terminals is:     $V = \varepsilon - Ir$

• Full Circuit Current: For an external resistor $R$ connected to the cell:     $V = IR \implies IR = \varepsilon - Ir$     $I = \frac{\varepsilon}{R + r}$

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

Combination of Cells

• Series Combination: For $n$ cells:

  • $\varepsilon_{eq} = \varepsilon_1 + \varepsilon_2 + ... + \varepsilon_n$

  • $r_{eq} = r_1 + r_2 + ... + r_n$

  • If a cell is connected with reverse polarity, its $\varepsilon$ enters with a negative sign.

• Parallel Combination: For cells connected across common points:

  • $\frac{1}{r_{eq}} = \sum_{i=1}^{n} \frac{1}{r_i}$

  • $\frac{\varepsilon_{eq}}{r_{eq}} = \sum_{i=1}^{n} \frac{\varepsilon_i}{r_i}$

  • For two cells in parallel: $\varepsilon_{eq} = \frac{\varepsilon_1 r_2 + \varepsilon_2 r_1}{r_1 + r_2}$, $r_{eq} = \frac{r_1 r_2}{r_1 + r_2}$.

Kirchhoff’s Rules

• Junction Rule (Kirchhoff’s First Rule): At any junction, the sum of currents entering equals the sum of currents leaving. This is based on the conservation of charge.

• Loop Rule (Kirchhoff’s Second Rule): The algebraic sum of changes in potential around any closed loop is zero. This is based on the conservation of energy.

• Sign Conventions: Potential decrease occurs when moving across a resistor in the direction of current ($-IR$) and when moving from positive to negative terminal of a cell ($-\varepsilon$).

Wheatstone Bridge

• Structure: Four resistors $R_1, R_2, R_3, R_4$ arranged in a bridge. A source is connected across one diagonal (AC), and a galvanometer (G) across the other (BD).

• Balanced Condition: When no current flows through the galvanometer ($I_g = 0$):     $\frac{R_1}{R_2} = \frac{R_3}{R_4}$

• Application: Used to find unknown resistance. If $R_4$ is unknown, and the bridge is balanced by varying $R_3$:     $R_4 = R_3 \frac{R_2}{R_1}$

Examples and Calculations

• Example 3.3 (Nichrome Toaster):

  • $R_{27} = 75.3\,\Omega$, $V = 230\,V$, $I_{steady} = 2.68\,A$

  • $R_{steady} = 230 / 2.68 = 85.8\,\Omega$

  • Using $R_2 = R_1 [1 + \alpha(T_2 - T_1)]$ with $\alpha = 1.70 \times 10^{-4}\,^{\circ}C^{-1}$

  • $T_2 - 27 = \frac{85.8 - 75.3}{75.3 \times 1.70 \times 10^{-4}} = 820\,^{\circ}C$.

  • Steady temperature $T_2 = 847\,^{\circ}C$.

• Example 3.4 (Platinum Thermometer):

  • $R_0 = 5\,\Omega$, $R_{100} = 5.23\,\Omega$, $R_t = 5.795\,\Omega$

  • $t = \frac{R_t - R_0}{R_{100} - R_0} \times 100 = \frac{0.795}{0.23} \times 100 = 345.65\,^{\circ}C$.

• Example 3.5 (Cubical Network):

  • 12 resistors of $1\,\Omega$ each in a cube. Battery $10\,V$.

  • Equivalent resistance $R_{eq} = \frac{5}{6}R = \frac{5}{6}\,\Omega$.

  • Total current $3I = \frac{10}{5/6} = 12\,A$. Therefore, $I = 4\,A$.

Points to Ponder

• Current as Scalar: Although drawn with arrows, current is a scalar because it follows the algebraic sum, not vector addition. It is the scalar product of current density and area vectors ($I = \mathbf{j} \cdot \Delta\mathbf{S}$).

• Resistance Meaning: The equation $V = IR$ defines resistance for any device; Ohm's law specifically states that $R$ is independent of $V$ (constant slope).

• Drift Complexity: Drift velocity is only due to the electric field; effects of random collisions average to zero.

• Charge Neutrality: In a neutral wire carrying current, the charge density $\rho$ is zero, even though current density $\mathbf{j}$ is non-zero.