LOADING REVISION
CHAPTER 03

Current Electricity
Quick Revision

High-yield formulas and concepts for rapid exam preparation.

Current Electricity

Electric Current

It is a scalar quantity defined as the rate of flow of electric charge through an area in the time interval.

I = q / t
  • SI Unit: Ampere (A)
  • It is a scalar quantity.

Ohm's Law

Current flowing through a circuit is directly proportional to the voltage applied to it.

I ∝ V ⇒ V = IR  (R = Resistance Ω)

Limitations of Ohm's Law

  • Only applicable to ohmic conductors.
  • Accuracy can be affected by temperature change and low currents.
  • There can be more than one value of V for the same current I. GaAs exhibits same type of behaviour.

Temperature dependence of Resistivity

ρ = ρ₀ [1 + α(T - T₀)]
  • ρ: Resistivity at temperature T
  • ρ₀: Resistivity at temperature T₀
  • α: Temperature coefficient of resistivity

Electric Cell

It is a device which maintains a potential difference between two terminals.

EMF

The work done per unit charge by the source in taking the charge from lower to higher potential energy is called electromotive force.

Internal Resistance

V_A - V_B = E - ir

r = Internal resistance of the cell

Power

The energy dissipated per unit time is known as power.

P = I²R = V² / R = VI

Drift Velocity

It is the average velocity, which is independent of time and is due to electric field.

v_d = -(eE / m) τ   |   v_d = I / (neA)
Vector form: J = ne v_d = σE   |   σ = (ne²τ) / m
Mobility: μ = v_d / E = eτ / m
  • τ: average collision time for electrons
  • Order of drift velocity of electrons ~ 10⁻⁴ ms⁻¹

Kirchhoff's Law

Current Law (Junction Law)

It states that algebraic sum of all the currents directed towards a point (junction) is zero.

i₁ + i₂ + i₄ = i₃ + i₅ ⇒ ΣI = 0

Voltage Law (Loop Law)

The algebraic sum of all the potential difference along a closed loop is zero.

ΣΔV = 0

Wheatstone Bridge

To measure unknown resistance in electrical circuits.

  • If R₁/R₂ = R₃/R₄, it is known as balance condition for galvanometer to give zero or null deflection.
  • No current will flow through S and V_C = V_D.

Current Electricity Fundamentals

1. Current Electricity

Detailed Concept: The study of electric charges in motion is broadly classified as current electricity. Unlike electrostatics which deals with stationary charges, this branch analyzes continuous charge transfer, current carriers, circuits, and electrical energy dissipation.

  • Core Focus: Dynamics of charge flow in conductors, potential differences, and circuit laws.
  • Practical Relevance: Powers all electronic devices, household wiring, and electrical machinery.

2. Electric Current

Detailed Concept: The systematic flow of electric charges through a given cross-section of a conductor constitutes an electric current. Quantitatively, it measures the volume of charge crossing a perpendicular area per unit time.

For steady flow: I = Q / t
For variable flow: I = lim (ΔQ / Δt) = dQ / dt

Units & Scale Definition

1 ampere = 1 coulomb / 1 second  ⇒  1 A = 1 C s⁻¹
  • SI Unit: Ampere (A), named after André-Marie Ampère.
  • Scalar Nature: Although current has both magnitude and a directional arrow indicating flow, it is a scalar quantity because electric currents do not obey the vector addition laws (like the parallelogram law). Instead, they follow simple algebraic addition.

3. Conventional and Electronic Currents

Detailed Concept: Directional analysis of charge carriers inside circuits involves two perspectives:

  • Conventional Current: Historically assumed direction of motion of positive charge carriers. It flows externally from the positive terminal to the negative terminal of a cell.
  • Electronic Current: Since electrons are negatively charged subatomic particles, their actual physical drift occurs in the opposite direction—flowing from the negative terminal to the positive terminal.

4. Electromotive Force (EMF)

Detailed Concept: The total work done by a non-electrostatic source (like a chemical cell or generator) in carrying a unit positive charge completely around the entire circuit loop (including inside the source itself).

EMF (E) = Work Done / Charge  ⇒  E = W / q

Detailed Breakdown & Volts

It is numerically equal to the terminal potential difference measured across the cell terminals when no current is drawn in an open circuit.

  • SI Unit: Volt (V).
  • Condition: If an electrochemical cell supplies 1 joule of energy to move 1 coulomb of charge through the complete circuit, its emf is exactly 1 volt.

5. Ohm's Law

Detailed Concept: Formulated by Georg Simon Ohm, this fundamental law establishes that the current flowing through a metallic conductor is directly proportional to the potential difference across its extreme ends, provided physical conditions (temperature, mechanical strain) remain entirely constant.

V ∝ I   ⇒   V = RI   ⇒   V / I = R

Resistance Parameter

Here, constant R is termed the electrical resistance of the conductor, acting as an opposition metric against charge carriers.

6. Resistance & Resistivity

Detailed Concept: Resistance is the intrinsic property of a conductor by virtue of which it resists or opposes the steady flow of electric charges. It depends directly on length and inversely on cross-sectional area.

R = ρ (l / A)   [where ρ = resistivity of material]

SI Unit and Definition of Ohm

1 ohm = 1 volt / 1 ampere  ⇒  1 Ω = 1 V A⁻¹

The resistance of a conductor is defined as 1 ohm if a current of 1 ampere flows through it when a potential difference of 1 volt is applied across its ends.

Resistivity, Current Density & Conduction Dynamics

1. Resistivity or Specific Resistance

Detailed Concept: Resistivity is defined as the characteristic resistance offered by a unit cube of a material of a conductor when current passes perpendicular to opposite faces. Unlike absolute resistance, resistivity is independent of geometrical dimensions (length or area) and depends entirely on the material's internal atomic structure.

ρ = (R × A) / l

Units & Factors

SI unit of ρ = Ω m (Ohm-meter)
  • Material Dependence: Depends directly on the nature/type of the conductor material.
  • Physical Conditions: Varies significantly with external physical parameters like temperature and mechanical pressure.

2. Current Density (j)

Detailed Concept: Current density represents the amount of electric charge flowing per second per unit cross-sectional area held strictly normal (perpendicular) to the direction of charge flow. It is a fundamental vector quantity pointing in the identical direction of motion of positive charge carriers.

For normal flow: j = (q / t) / A = I / A
General Vector Form: I = j A cos θ ⇒ I = j · A

SI Unit

SI unit of current density = A m⁻²

3. Conductance (G)

Detailed Concept: Conductance is an ease-of-flow metric, representing the exact reciprocal of electrical resistance. It measures how readily a material allows electric current to pass through its circuit paths.

Conductance = 1 / Resistance ⇒ G = 1 / R

Units & Alternate Names

SI unit = ohm⁻¹ = mho = siemen (S)

4. Conductivity or Specific Conductance (σ)

Detailed Concept: Conductivity is the reciprocal of material resistivity, quantifying a substance's capability to conduct electricity under an electric field gradient.

Conductivity = 1 / Resistivity ⇒ σ = 1 / ρ

SI Units

SI unit = ohm⁻¹ m⁻¹ = mho m⁻¹ = S m⁻¹ (Siemens per meter)

5. Resistivities Classification

Detailed Concept: Materials are classified into three major groups based on their resistivity ranges:

  • Metals (Conductors): Possess extremely low resistivities ranging from $10^{-8}\ \Omega\text{ m}$ to $10^{-6}\ \Omega\text{ m}$, ensuring high free electron mobility.
  • Insulators: Exhibit very high resistivities greater than $10^{4}\ \Omega\text{ m}$, restricting any free charge movement.
  • Semiconductors: Display intermediate resistivities lying between $10^{-6}\ \Omega\text{ m}$ and $10^{4}\ \Omega\text{ m}$ (e.g., Germanium, Silicon).

6. Charge Carriers of Current

Detailed Concept: Electric current is carried by different mobile charge carriers depending on the physical state and chemical nature of the medium:

  • Metals: Free valence electrons act as the primary mobile charge carriers.
  • Ionised Gases: Free electrons and positively charged ions constitute the carriers.
  • Electrolytes: Both positive ions (cations) and negative ions (anions) transport charge.
  • Semiconductors (Ge, Si): Conduction happens via dual carriers—free electrons and holes. A hole represents a vacant electronic state left behind when an electron is removed, behaving effectively as a positive charge carrier.

7. Drift Velocity & Relaxation Time

Detailed Concept: Drift velocity ($v_d$) is the small average net velocity acquired by free electrons inside a conductor when an external electric field is applied, driving them opposite to the field lines.

  • Relaxation Time ($\tau$): The average time elapsed between two successive electronic collisions inside the atomic lattice.
  • Magnitude: Thermal speeds are high, but the net drift velocity of electrons is very slow, roughly around $10^{-4}\text{ ms}^{-1}$.

Mobility & Current Relations (Detailed Breakdown)

5. Mobility of Charge Carriers (μ)

Detailed Concept: Mobility defines how easily and fast a charge carrier moves through a conductor when an external electric field intensity is applied. It is quantitatively measured as the drift velocity acquired per unit electric field.

μ = vd / E = (q τ) / m

Specific Carriers & Units

For Electrons: μe = (e τe) / me   |   For Holes: μh = (e τh) / mh
  • SI Unit: m2 V-1 s-1
  • Practical Unit: cm2 V-1 s-1

6. Current & Conductivity with Mobility

Detailed Concept: Formulating the total bulk current and net material conductivity by scaling individual carrier densities with their respective mobilities:

For Conductor: I = n e A μe E
For Semiconductor: I = e A E (n μe + p μh)
Conductivity: σ = e (n μe + p μh)

Variables Breakdown

Here, n and p represent the free electron concentration and hole concentration per unit volume respectively.

Resistances, Cell EMF & Terminal Voltage Dynamics

1. Resistances in Series

Mathematical Concept: When multiple resistors are connected end-to-end such that the same electric current flows through each of them, their equivalent (total) resistance is equal to the direct algebraic sum of individual resistances.

Rs = R1 + R2 + R3 + ...

Symbolic Breakdown (Notations)

  • Rs: Equivalent or total resistance in Series combination.
  • R1, R2, R3: Individual resistances connected sequentially in the circuit line.

2. Resistances in Parallel

Mathematical Concept: When resistors are connected across two common points (nodes) so that the potential difference across each is the same, the reciprocal of the equivalent resistance equals the sum of reciprocals of individual resistances.

1 / Rp = (1 / R1) + (1 / R2) + (1 / R3) + ...
For two resistors: Rp = (R1 × R2) / (R1 + R2)

Symbolic Breakdown (Notations)

  • Rp: Equivalent or total resistance in Parallel combination.
  • R1, R2: Individual branch resistances sharing the same voltage drop.

3. Current Division in Parallel Circuits

Mathematical Concept: Total current entering a parallel network divides among the parallel branches in the inverse ratio of their resistances (i.e., smaller resistance gets larger current).

I1 = [R2 / (R1 + R2)] × I
I2 = [R1 / (R1 + R2)] × I

Symbolic Breakdown (Notations)

  • I: Total main-line current entering the parallel junction.
  • I1, I2: Branch currents flowing through resistors R1 and R2 respectively.
  • R1, R2: Resistor values in the parallel branches.

4. Internal Resistance (r)

Mathematical Concept: The obstruction or resistance offered by the electrolytic solution and electrodes of a cell to the internal flow of ionic current is termed internal resistance (r).

Factors Affecting Internal Resistance (r)

  • (i) Nature of electrolyte: Higher ionic viscosity increases r.
  • (ii) Concentration: Higher concentration raises r.
  • (iii) Distance between electrodes: Greater separation increases r.
  • (iv) Common area: Larger submerged electrode area decreases r.
  • (v) Temperature: Raising electrolyte temperature decreases r.

5. EMF (ℰ) vs Terminal Voltage (V)

Mathematical Concept: When a cell is discharging in a closed circuit, its terminal voltage drops below its electromotive force (EMF) due to potential drop across its internal resistance.

During Discharging:   ℰ = V + I·r   ⇒   V = ℰ - I·r
Ohmic Form:   V = I·R = [ℰ·R] / [R + r]

Determining Internal Resistance (r) & Current Formulas

r = (ℰ - V) / I   =οr=   r = R · [(ℰ - V) / V]
Circuit Current: I = ℰ / (R + r)   |   Max Current: Imax = ℰ / r

6. Cell Charging Condition

Mathematical Concept: When an external source forces current backward into a secondary cell to recharge it, current enters its positive terminal, causing the terminal voltage to exceed its EMF.

During Charging:   V = ℰ + I·r   (where V > ℰ)

Complete Variable Glossary (Master Legend)

  • ℰ (EMF): Electromotive force; max open-circuit potential difference across cell terminals.
  • V: Terminal potential difference across the load resistor when current flows.
  • I: Total electric current circulating through the loop.
  • r: Internal resistance belonging inside the electrochemical cell.
  • R: External load resistance connected across the cell.
  • Imax: Maximum possible current drawn when external resistance R = 0 (short circuit).

Cells Combination & Joule's Heating Effect (Simplified Guide)

1. Cells in Series Grouping

Simple Explanation: When multiple identical cells are connected one after another (positive terminal of one connected to negative of the next) in a single path with an external load resistor, their total voltage (EMF) adds up.

I = (n ℰ) / (R + n r)

Sign / Notation Glossary (क्या-क्या है इसमें?)

  • I: Total current flowing through the external circuit (अम्पीयर में कुल धारा).
  • n: Total number of cells connected in series line (श्रेणी में जुड़े सेलों की कुल संख्या).
  • ℰ (mathcal{E}): EMF of each individual cell (प्रत्येक सेल का विद्युत वाहक बल).
  • R: External load resistance connected in the circuit (बाहरी प्रतिरोध).
  • r: Internal resistance of each single cell (प्रत्येक सेल का आंतरिक प्रतिरोध).

2. Cells in Parallel Grouping

Simple Explanation: When all positive terminals of multiple cells are joined together at one common point and all negative terminals at another, they are in parallel. This helps provide higher current stability.

I = (m ℰ) / (m R + r)

Sign / Notation Glossary (क्या-क्या है इसमें?)

  • I: Total circuit current.
  • m: Total number of rows/cells connected in parallel (समानांतर शाखाओं की संख्या).
  • ℰ: EMF of each cell.
  • R: External resistance.
  • r: Internal resistance of each cell.

3. Cells in Mixed Grouping

Simple Explanation: A combination where multiple cells are first connected in series in several rows, and all those rows are then connected in parallel with each other to get maximum efficiency.

I = (m n ℰ) / (m R + n r)
For Maximum Current: R = (n r) / m  ⇒  m R = n r

Sign / Notation Glossary & Condition

  • n: Number of cells connected in series per row.
  • m: Number of parallel rows.
  • Condition for Max Current: External resistance (R) must equal total internal resistance of the battery setup.

4. Heating Effect (Joule Heating)

Simple Explanation: When electric current passes through a resistor, electrical energy is converted into heat energy due to collisions of moving electrons with lattice ions. This is an irreversible process.

Key Features

  • It is completely irreversible (heat produced cannot automatically turn back into full electricity).
  • Found in everyday items like electric heaters, irons, and geysers.

5. Joule's Law of Heating & Formulas

Mathematical Concept: The amount of heat ($H$) produced in a resistor depends directly on current square, resistance value, and flow time.

H = V I t  =  I² R t  =  (V² t) / R   (in Joules)
In Calories:   H = (V I t) / 4.18  =  (I² R t) / 4.18  =  (V² t) / (4.18 R)

Sign / Notation Glossary (हर एक संकेत का मतलब)

  • H: Total heat energy produced (उत्पन्न ऊष्मा).
  • V: Potential difference across the resistor (विभवांतर).
  • I: Electric current flowing through the circuit (धारा).
  • R: Resistance of the resistor (प्रतिरोध).
  • t: Time duration for which current flows (समय सेकंड में).
  • 4.18: Joule's mechanical equivalent of heat constant (1 calorie = 4.18 Joules).

Electric Power, Energy, Efficiency & Maximum Power Theorem

1. Electric Power (विद्युत शक्ति)

Simple Explanation: जिस दर से कोई विद्युत उपकरण (electrical appliance) विद्युत ऊर्जा को दूसरी ऊर्जा में बदलता है, या जिस दर से बैटरी/सेल सर्किट में करंट बनाए रखने के लिए काम करता है, उसे पावर कहते हैं।

P = W / t  =  V · I  =  I² · R  =  V² / R

Sign & Symbol Glossary (हर चिन्ह का मतलब समझें)

  • P: Electric Power / विद्युत शक्ति (वाट में)।
  • W: Work Done or Electrical Energy / किया गया कुल कार्य या ऊर्जा (जूल में)।
  • t: Time / समय (सेकंड में)।
  • V: Potential Difference / विभवांतर या वोल्टेज (वोल्ट में)।
  • I: Electric Current / विद्युत धारा (एम्पियर में)।
  • R: Resistance / प्रतिरोध (ओम में)।

2. SI Unit of Power (वाट)

Simple Explanation: पावर की यूनिट वाट (Watt) होती है। यदि किसी उपकरण में 1 वोल्ट विभवांतर लगाने पर 1 एम्पियर धारा बहे, तो उसकी पावर 1 वाट होगी।

1 watt = 1 joule / 1 second  ⇒  1 W = 1 J s⁻¹ = 1 V A
Commercial/Bigger Unit:   1 kilowatt (kW) = 1000 W

3. Electric Energy (विद्युत ऊर्जा)

Simple Explanation: किसी दिए गए समय में सर्किट में विद्युत धारा को बनाए रखने के लिए किया गया कुल कार्य ही विद्युत ऊर्जा है।

Electric Energy (W) = P · t  =  V · I · t  =  I² · R · t  =  (V² · t) / R

Commercial Units (व्यापारिक मात्रक)

  • Kilowatt-hour (kWh / B.O.T. Unit): इसे ही 'यूनिट' कहते हैं (1 kWh = 1000 W बिजली 1 घंटे चलने पर खर्च ऊर्जा)।
  • Calculation: $1\text{ kWh} = 1000\text{ W} \times 3600\text{ s} = 3.6 \times 10^6\text{ Joules}$.

4. Power Rating & Combination (उपकरणों की पावर)

Simple Explanation: किसी उपकरण पर लिखा पावर उसका रेटेड पावर होता है जो वह तय वोल्टेज पर खपत करता है। जब कई उपकरण जोड़े जाते हैं:

Series Combination (श्रेणी क्रम):   1 / P = (1 / P₁) + (1 / P₂) + (1 / P₃) + ...
Parallel Combination (समानांतर क्रम):   P = P₁ + P₂ + P₃ + ...

Symbolic Glossary

  • P₁, P₂, P₃: अलग-अलग उपकरणों की व्यक्तिगत पावर रेटिंग (Watt में)।

5. Efficiency of a Cell/Source ($\eta$)

Simple Explanation: किसी सेल द्वारा दी गई कुल पावर (इनपुट) में से कितनी पावर बाहरी लोड को मिलती है (आउटपुट), उसका अनुपात दक्षता ($\eta$) कहलाता है।

η = Output Power / Input Power  =  (V · I) / (ℰ · I)  =  V / ℰ  =  R / (R + r)

Sign & Symbol Glossary

  • η (Eta): Efficiency / दक्षता (प्रतिशत या अनुपात में)।
  • ℰ (mathcal{E}): EMF of source / सेल का विद्युत वाहक बल।
  • V: Terminal Voltage / सिरों के बीच का विभवांतर।
  • R: External Resistance / बाहरी प्रतिरोध।
  • r: Internal Resistance / सेल का आंतरिक प्रतिरोध।

6. Maximum Power Theorem

Simple Explanation: किसी सोर्स से अधिकतम आउटपुट पावर तब मिलती है जब सर्किट का बाहरी प्रतिरोध (R) सेल के आंतरिक प्रतिरोध (r) के बिल्कुल बराबर हो जाता है ($R = r$)।

Condition: R = r   |   Maximum Power: Pmax = ℰ² / (4r)

Special Point

जब सर्किट में अधिकतम पावर डिलीवर हो रही होती है, तब उस समय सेल की दक्षता ($\eta$) ठीक 50% होती है।

7. Joule's Law of Heating (जूल का ऊष्मीय नियम)

Simple Explanation: जब किसी प्रतिरोधक से धारा बहती है, तो ऊर्जा ऊष्मा (Heat) के रूप में नष्ट होती है। यह एक अपरिवर्तनीय (irreversible) प्रक्रिया है।

H = V · I · t  =  I² · R · t  =  (V² · t) / R   (Joules में)
In Calories:   H = (I² · R · t) / 4.18  cal

Symbolic Glossary

  • H: Heat energy produced / उत्पन्न ऊष्मा।
  • 4.18: Joule's mechanical equivalent constant (1 cal = 4.18 J).

8. Efficiency of Electric Motor

Simple Explanation: मोटर में दी गई कुल विद्युत ऊर्जा का कितना हिस्सा यांत्रिक कार्य (Mechanical Work) में बदलता है।

η = Output Mechanical Power / Input Electric Power
Input Power = Output Mechanical Power + Loss as Heat

Important Condition

किसी इलेक्ट्रिक मोटर की आउटपुट पावर अधिकतम तब होती है जब उसकी बैक-ईएमएफ (Back EMF), सोर्स ईएमएफ की ठीक आधी होती है (शर्त यह है कि मोटर की कॉइल का प्रतिरोध नगण्य हो)।

Kirchhoff's Laws, Potentiometer & Wheatstone Bridge Dynamics

1. Kirchhoff's First Law (Junction Rule)

Detailed Concept: In an electric circuit, the algebraic sum of currents meeting at any junction is strictly zero. Alternatively, the total sum of currents entering a junction equals the total sum of currents leaving that junction.

Mathematical Expression:   ΣI = 0

Fundamental Justification & Notations

This law is founded upon the law of conservation of charge. When currents in a circuit are steady, electric charges cannot accumulate or originate at any point inside the circuit.

  • I: Electric current flowing through circuit branches (measured in Amperes, A).

2. Kirchhoff's Second Law (Loop Rule)

Detailed Concept: Around any closed loop of a network, the algebraic sum of changes in potential must be zero. Or, the algebraic sum of EMFs in any closed loop equals the sum of the products of currents and resistances in it.

ΣΔV = 0   |   Σℰ = Σ(I · R)

Fundamental Justification & Notations

This law is based on the law of conservation of energy. Since the electrostatic force field is conservative, the total work done by it along any closed path is identically zero.

  • ℰ (mathcal{E}): Electromotive force of source cells.
  • I: Branch currents.
  • R: Resistor values in respective branches.

3. Galvanometer

Detailed Concept: A highly sensitive electromechanical instrument utilized to detect the presence and direction of minute electric currents in a circuit.

Working Characteristic

It produces a mechanical pointer deflection directly proportional to the magnitude of the electric current flowing through its coil.

4. Potentiometer & Principle

Detailed Concept: A versatile laboratory device used primarily to compare the electromotive forces (EMFs) of two electrical sources accurately without drawing current.

Principle:   V ∝ l  ⇒  V = k · l

Principle & Notations

When a steady current flows through a uniform resistance wire, the potential drop across any length is directly proportional to that length.

  • V: Potential difference across wire length.
  • l: Balancing length of the wire.
  • k: Potential drop per unit length, known as the potential gradient.

5. Applications of Potentiometer

Detailed Concept: Two primary practical uses of a potentiometer setup:

(i) Comparison of Two Cell EMFs

ℰ₁ / ℰ₂ = l₁ / l₂

Here, l₁ and l₂ are balancing lengths for source EMFs ℰ₁ and ℰ₂.

(ii) Finding Internal Resistance (r) of a Cell

r = [(l₁ - l₂) / l₂] · R

Notations: l₁ = balancing length without external shunt; l₂ = balancing length with shunt resistance R across the cell.

6. Wheatstone Bridge

Detailed Concept: An electrical bridge arrangement consisting of four resistive arms (P, Q, R, S) joined to form quadrilateral ABCD, with a battery across diagonal AC and a sensitive galvanometer across BD.

Balanced Condition:   P / Q = R / S

Operational Details & Notations

Resistances are adjusted until zero current flows through the galvanometer (null deflection). Knowing any three resistors allows precise calculation of the fourth. Maximum sensitivity is achieved when all four bridge arms have comparable resistance magnitudes.

  • P, Q, R, S: Bridge arm resistances.

7. Slide Wire Bridge (Metre Bridge)

Detailed Concept: A practical engineering realization of a Wheatstone bridge where wire length is 100 cm. Resistance R is fixed, and the balance point is located by sliding a jockey.

P / Q = R / S = l / (100 - l)   ⇒   S = [(100 - l) / l] · R
Material Resistivity:   ρ = (S · A) / l_wire = (S · π · r²) / l

Variable Glossary (Symbols Explained)

  • l: Balancing length measured from the left end (in cm).
  • (100 - l): Remaining length of the metre bridge wire.
  • S: Unknown resistance to be measured.
  • A: Cross-sectional area of wire ($\pi r^2$).
  • ρ (rho): Material resistivity.

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