Current Electricity

Introduction to Electric Currents and Charge Motion

  • Charges in motion constitute an electric current, whereas charges at rest were the focus of previous electrostatic studies.

  • Electric currents occur in both natural and artificial forms:

    • Natural Phenomena: Lightening is a prominent example where charges flow from clouds to the Earth through the atmosphere. The current produced during lightening is not steady.

    • Artificial Generation: Currents are generated using dynamos, inverters, and cells. Devices such as cell-driven clocks and torches utilize steady electric currents.

Fundamental Definition of Electric Current

  • Consider a small area AA kept perpendicular to the direction of the flow of charges.

  • If positive charges q+q_{+} flow from left to right and negative charges q−q_{-} flow from right to left across the area, the net charge qq flowing through the area in time interval tt from left to right is defined as:   q=q+−q−q = q_{+} - q_{-}

  • For a steady current, the charge qq is proportional to time tt.

  • The quotient I=qtI = \frac{q}{t} defines the current across the area in the direction left to right. A negative quotient implies current flow from right to left.

  • Non-Steady Currents: When currents are not steady, two additional definitions are used:

    1. Average Current (IavI_{av}): If a charge △Q\triangle Q flows through a cross-section in the interval from tt to t+△tt + \triangle t, then:        Iav=△Q△tI_{av} = \frac{\triangle Q}{\triangle t}

    2. Instantaneous Current (II): Defined as the limit of the average current as the time interval approaches zero:        I=lim△t→0△Q△t=dQdtI = \text{lim}_{\triangle t \rightarrow 0} \frac{\triangle Q}{\triangle t} = \frac{dQ}{dt}

  • Direction and Units:

    • The direction of current is traditionally taken as the direction of motion of positively charged particles and opposite to the direction of motion of negatively charged particles.

    • The SI unit of current is the Ampere (A\text{A}).

  • Orders of Magnitude:

    • Domestic appliances: 1 A\text{1 A}

    • Human nerves: 10−6A10^{-6} \text{A}

    • Lightening: 104A10^{4} \text{A}

Electric Current in Conductors and Mobility

  • Force and Motion: When an electric field is applied to a charge, it experiences a force. If the charge is free to move, it contributes to a current.

  • Mobility (ν\nu): This is defined as the magnitude of drift velocity per unit electric field:   ν=∣vd∣E\nu = \frac{|v_{d}|}{E}

  • SI Unit of Mobility: m2V−1s−1\text{m}^{2} \text{V}^{-1} \text{s}^{-1}.

  • Practical Unit of Mobility: cm2V−1s−1\text{cm}^{2} \text{V}^{-1} \text{s}^{-1}.

  • Given that drift velocity vd=eEτmv_{d} = \frac{e E \tau}{m}, the mobility can be expressed as:   ν=eτm\nu = \frac{e \tau}{m}

  • Mobility of free electrons is independent of the electric field and the dimensions of the conductor.

Ohm's Law and Resistance

  • Basic Proportionality: Under constant physical conditions (like temperature and mechanical strain), the current flowing through a metallic conductor is directly proportional to the potential difference across its ends:   V×I→V=IRV \times I \rightarrow V = IR

  • Resistance (RR): The constant of proportionality is called resistance, which depends on the material and dimensions of the conductor.

  • Resistivity (ρ\rho): Resistance is proportional to length (ll) and inversely proportional to the area of cross-section (AA):   R×lA→R=ρlAR \times \frac{l}{A} \rightarrow R = \rho \frac{l}{A}

  • The constant ρ\rho is the resistivity or specific resistance. It depends on the material but is independent of its dimensions.

  • Conductivity (τ\tau): The reciprocal of resistivity:   τ=1ρ\tau = \frac{1}{\rho}

  • Current Density and Electric Field:

    • Current density (jj) is current per unit normal area: j=IAj = \frac{I}{A}.

    • Vectorially: I = \text{j} \times \text{A} = \text{\int} \text{j} \times d\text{A}.

    • Relation to Electric Field: E=jρE = j \rho or vectorially E=jρ\text{E} = \text{j} \rho.

    • Equivalent Ohm's Law: j=τE\text{j} = \tau \text{E}.

Dimensional Changes and Resistance

  • When volume or mass of a conductor remains constant, the resistance varies based on changes in dimensions:

    1. If length becomes nn times, resistance becomes n2n^2 times the initial value (since R×L2R \times L^2).

    2. If area becomes nn times, resistance becomes 1n2\frac{1}{n^2} times the initial value (since R×1A2R \times \frac{1}{A^2}).

    3. If length increases by x \text{%} (where x < 5), resistance increases by 2x \text{%}.

    4. If area increases by x \text{%} (where x < 5), resistance decreases by 2x \text{%}.

    5. If the radius of the cross-section increases by x \text{%} (where x < 5), resistance decreases by 4x \text{%}.

Drift Velocity and the Origin of Resistivity

  • Absence of Electric Field: Free electrons collide with fixed ions and emerge in random directions with the same speed. The average velocity of NN free electrons is zero:   \nu_{avg} = \frac{1}{N} \text{\sum}_{i=1}^{N} \text{u}_{i} = 0

  • Presence of Electric Field (EE): Each electron experiences an acceleration a=−eEma = \frac{-e E}{m}.

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

  • Drift Velocity (vdv_{d}): The velocity after time tit_{i} since the last collision:   νi=ui−eEmti\nu_{i} = \text{u}_{i} - \frac{e E}{m} t_{i}

  • Average drift velocity is independent of time despite acceleration:   vd=−eEmτv_{d} = -\frac{e E}{m} \tau

  • Current Relation: The total charge transported in time △t\triangle t is △Q=neA∣vd∣△t\triangle Q = n e A |v_{d}| \triangle t.

  • This leads to the current formula: I=ne2AτmEI = \frac{n e^2 A \tau}{m} E.

  • From this, conductivity is derived as τ=ne2τm\tau = \frac{n e^2 \tau}{m}, where nn is the number of free electrons per unit volume.

Material Resistivity and Resistors

  • Resistivity Ranges at 0∘C0^{\circ}\text{C}:

    • Silver: 1.6×10−8 Ωm1.6 \times 10^{-8} \text{ Ωm}, α=0.0041 °C−1\text{α} = 0.0041 \text{ °C}^{-1}

    • Copper: 1.7×10−8 Ωm1.7 \times 10^{-8} \text{ Ωm}, α=0.0068 °C−1\text{α} = 0.0068 \text{ °C}^{-1}

    • Aluminium: 2.7×10−8 Ωm2.7 \times 10^{-8} \text{ Ωm}, α=0.0043 °C−1\text{α} = 0.0043 \text{ °C}^{-1}

    • Nichrome: ≈100×10−8 Ωm\text{≈} 100 \times 10^{-8} \text{ Ωm}, α=0.0004 °C−1\text{α} = 0.0004 \text{ °C}^{-1}

    • Semiconductors (e.g., Silicon): 2300 Ωm2300 \text{ Ωm}, α=−0.07 °C−1\text{α} = -0.07 \text{ °C}^{-1}

    • Insulators (e.g., Fused Quartz): 1014−1016 Ωm10^{14} - 10^{16} \text{ Ωm}

  • Types of Resistors:

    1. Wire Bound Resistors: Made by winding wires of alloys like Manganin, Constantan, or Nichrome. They are chosen because their resistivities are relatively insensitive to temperature. Range is fraction of an ohm to a few hundred ohms.

    2. Carbon Resistors: Small, compact, and inexpensive. Used widely in electronic circuits. Values are determined via a color code.

Resistor colour Codes

  • A carbon resistor has coaxial colored rings. The first two bands represent the first two significant figures. The third band is the decimal multiplier. The fourth band indicates tolerance.

  • Color numbering and multiplier:

    • Black: 0, 11

    • Brown: 1, 10110^{1}

    • Red: 2, 10210^{2}

    • Orange: 3, 10310^{3}

    • Yellow: 4, 10410^{4}

    • Green: 5, 10510^{5}

    • Blue: 6, 10610^{6}

    • Violet: 7, 10710^{7}

    • Gray: 8, 10810^{8}

    • White: 9, 10910^{9}

    • Gold: Multiplier 10−110^{-1}, Tolerance \text{±} 5\text{%}

    • Silver: Multiplier 10−210^{-2}, Tolerance \text{±} 10\text{%}

    • No color: Tolerance \text{±} 20\text{%}

Temperature Dependence of Resistivity

  • Resistivity is temperature-dependent: ρ=mne2τ\rho = \frac{m}{n e^2 \tau}.

  • Metals: As temperature increases, electron speed increases, collision frequency increases, and relaxation time τ\tau decreases. This causes resistivity to increase. The relation over a limited range is:   ρT=ρ0[1+α(T−T0)]\rho_{T} = \rho_{0} [1 + \text{α}(T - T_{0})]

  • Semiconductors: For these materials, α\text{α} is negative because the number of free electrons nn increases significantly with temperature, causing resistivity to decrease.

  • Resistance at temperature tt: Rt=R0(1+αRt)R_{t} = R_{0} (1 + \text{α}_{R} t).

  • Platinum Resistance Thermometer Principle:   t=Rt−R0R100−R0×100 °Ct = \frac{R_{t} - R_{0}}{R_{100} - R_{0}} \times 100 \text{ °C}

Limitations of Ohm's Law

  • Deviations from linearity occur in several cases:

    1. VV ceases to be proportional to II for good conductors at high currents.

    2. The value of current depends on the direction of potential difference (e.g., Diodes).

    3. The relationship between VV and II is not unique (e.g., Gallium Arsenide/GaA), where multiple values of potential may exist for the same current.

Combination of Resistors

  • Series Combination:

    • Identical current flows through all resistors.

    • Total potential: V=V1+V2=I(R1+R2)V = V_1 + V_2 = I(R_1 + R_2).

    • Equivalent resistance: Req=R1+R2+...+RnR_{eq} = R_1 + R_2 + \text{...} + R_{n}.

    • ReqR_{eq} is always greater than the greatest individual resistance.

  • Parallel Combination:

    • Potential difference across each resistor is the same.

    • Total current: I=I1+I2=V(1R1+1R2)I = I_1 + I_2 = V(\frac{1}{R_1} + \frac{1}{R_2}).

    • Equivalent resistance: 1Req=1R1+1R2+...+1Rn\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \text{...} + \frac{1}{R_n}.

    • For two resistors: Req=R1R2R1+R2R_{eq} = \frac{R_1 R_2}{R_1 + R_2}.

Electrical Energy and Power

  • When a charge △Q=I△t\triangle Q = I \triangle t moves across potential difference VV, potential energy changes by △U=−△QV=−IV△t\triangle U = - \triangle Q V = - I V \triangle t.

  • If the charge carrier gains no kinetic energy (steady drift velocity), this energy is dissipated as heat in the conductor.

  • Heat Disipated: △W=IV△t\triangle W = I V \triangle t.

  • Power (PP): Energy dissipated per unit time:   P=VI=I2R=V2RP = VI = I^2 R = \frac{V^2}{R}

  • Power Loss in Transmission Lines: To minimize power loss (Pc=P2RcV2P_c = \frac{P^2 R_c}{V^2}) over long distances (RcR_c), electricity is transmitted as low current at very high voltages using transformers.

Cells, EMF, and Internal Resistance

  • Electrolytic Cell: Consists of two electrodes (Positive PP and Negative NN) in an electrolyte.

  • Electromotive Force (ε): The potential difference between terminals when the circuit is open (I=0I=0):   ε=V++V−\text{ε} = V_{+} + V_{-}

  • Internal Resistance (rr): The finite resistance offered by the electrolyte.

  • Cell Voltage (VV):

    • Discharging: V=ε−IrV = \text{ε} - Ir.

    • Charging: V=ε+IrV = \text{ε} + Ir.

    • Open Circuit: V=εV = \text{ε}.

    • Ideal Cell: r=0→V=εr=0 \rightarrow V = \text{ε}.

  • Circuit Current: If connected to external resistance RR, then I=εR+rI = \frac{\text{ε}}{R + r}.

  • Maximum Current: Occurs when R=0R = 0, then Imax=εrI_{max} = \frac{\text{ε}}{r}.

Grouping of Cells

  • Cells in Series:

    • Equivalent EMF: εeq=ε1+ε2+...+εn\text{ε}_{eq} = \text{ε}_1 + \text{ε}_2 + \text{...} + \text{ε}_n.

    • Equivalent internal resistance: req=r1+r2+...+rnr_{eq} = r_1 + r_2 + \text{...} + r_n.

    • If one cell is reversed (wrongly connected): εeq=ε1−ε2\text{ε}_{eq} = \text{ε}_1 - \text{ε}_2.

    • For nn identical cells with mm wrongly connected: εeq=(n−2m)ε\text{ε}_{eq} = (n - 2m)\text{ε} and req=nrr_{eq} = nr.

  • Cells in Parallel (Two Cells):

    • εeq=ε1r2+ε2r1r1+r2\text{ε}_{eq} = \frac{\text{ε}_1 r_2 + \text{ε}_2 r_1}{r_1 + r_2}

    • 1req=1r1+1r2\frac{1}{r_{eq}} = \frac{1}{r_1} + \frac{1}{r_2}

    • For nn identical cells in parallel: εeq=ε\text{ε}_{eq} = \text{ε} and req=rnr_{eq} = \frac{r}{n}.

Kirchhoff's Rules

  • First Rule (Junction Rule): At any junction, the sum of currents entering equals the sum of currents leaving (ΣI=0\text{Σ} I = 0). This is based on the conservation of charge.

  • Second Rule (Loop Rule): The algebraic sum of changes in potential around any closed loop is zero (Σ△V=0\text{Σ} \triangle V = 0). This is based on the conservation of energy.

  • Sign Conventions:

    • Potential decrease when moving in the direction of current across a resistor: △V=−IR\triangle V = -IR.

    • Potential increase when moving across a cell from negative to positive terminal: △V=+ε\triangle V = +\text{ε}.

Wheatstone Bridge and Meter Bridge

  • Wheatstone Bridge: An arrangement of four resistors (R1,R2,R3,R4R_1, R_2, R_3, R_4). The "balance condition" yields zero current in the galvanometer (Ig=0I_g = 0):   R1R2=R3R4\frac{R_1}{R_2} = \frac{R_3}{R_4}

  • Meter Bridge: A practical application of the Wheatstone Bridge using a uniform 1-metre wire.

    • Let RR be the unknown resistance and SS be the standard resistance.

    • At balance point length ll:       RS=l100−l→R=Sl100−l\frac{R}{S} = \frac{l}{100 - l} \rightarrow R = S \frac{l}{100 - l}

    • Percentage error is minimized by keeping the balance point near the middle (approx. 50 cm50 \text{ cm}).

Atmospheric Electricity Note

  • Atmospheric electricity results from charge separation in clouds.

  • Lightening fields are ordered at 105 volt/metre10^{5} \text{ volt/metre}.

  • A flash averages four strokes, with total flash duration around 30 seconds30 \text{ seconds}.

  • Fair weather surface charge density at ground is negative, with an electric field of about 120 volt/metre120 \text{ volt/metre} directed downward.

Numerical Examples Summary

  • Net Charge through an area: For 101410^{14} protons and 101410^{14} electrons moving in opposite directions in 10 μs10 \text{ μs}, the net current calculation:   q=q+−q−=1.6×10−5 C−(−1.6×10−5 C)=3.2×10−5 Cq = q_{+} - q_{-} = 1.6 \times 10^{-5} \text{ C} - (-1.6 \times 10^{-5} \text{ C}) = 3.2 \times 10^{-5} \text{ C}   I=3.2×10−5 C10×10−6 s=3.2 AI = \frac{3.2 \times 10^{-5} \text{ C}}{10 \times 10^{-6} \text{ s}} = 3.2 \text{ A}.

  • Current in Neon Discharge Tube: If 2.9×10182.9 \times 10^{18} Ne+Ne^{+} ions move right and 1.2×10181.2 \times 10^{18} electrons move left per second:   I=(n++n−)et=(2.9×1018+1.2×1018)×1.6×10−191=0.66 A to the rightI = \frac{(n_{+} + n_{-})e}{t} = \frac{(2.9 \times 10^{18} + 1.2 \times 10^{18}) \times 1.6 \times 10^{-19}}{1} = 0.66 \text{ A to the right}.

  • Current in Bohr Model (Hydrogen Atom): Electron rotating in circular orbit, r=5×10−11 mr = 5 \times 10^{-11} \text{ m}, v=2.2×106 m/sv = 2.2 \times 10^6 \text{ m/s}. Current associated with motion:   I=eT=ev2πr=1.6×10−19×2.2×1062×3.14×5×10−11≈1.1×10−3 AI = \frac{e}{T} = \frac{e v}{2 \text{π} r} = \frac{1.6 \times 10^{-19} \times 2.2 \times 10^6}{2 \times 3.14 \times 5 \times 10^{-11}} \text{≈} 1.1 \times 10^{-3} \text{ A}.