📘 Current Electricity
Physics • Class XII
Introduction to Electric Currents
Electric currents are the flow of charges, which can occur naturally, such as in lightning, or in steady forms in devices like torches and clocks. This chapter focuses on the basic laws concerning steady electric currents.
Definition of Electric Current
Electric current is defined as the net amount of charge flowing across an area in a given time interval. For steady current, the formula is: q = q+ - q- I = q/t where I is the current, q is the net charge, and t is the time interval. If I is negative, it indicates a current in the backward direction. In general, current is defined as: I(t) = lim (ΔQ/Δt) as Δt → 0 The SI unit of current is the ampere (A), which is significant in both domestic appliances and natural phenomena like lightning.
Electric Currents in Conductors
Electric charges experience a force in an electric field, leading to movement and current. Free charged particles exist in nature, but in atoms, electrons are usually bound to nuclei. - Conductors: Materials where some electrons are free to move, allowing electric currents to develop when an electric field is applied. - Solid conductors: Current is carried by negatively charged electrons among fixed positive ions. - Electrolytic solutions: Both positive and negative charges can move, but this chapter focuses on solid conductors.
Behavior of Electrons in Conductors
In the absence of an electric field, electrons move due to thermal motion and collide with each other. This behavior is crucial for understanding how currents are established in conductive materials. Electrons colliding with fixed ions in a conductor emerge with the same speed but in random directions. At any moment, the number of electrons moving in one direction equals those moving in the opposite direction, resulting in no net electric current.
Effect of Electric Field on Conductors
When an electric field is applied to a conductor (e.g., a cylindrical shape), it creates a directional force on electrons. Electrons are accelerated towards the positive charge, temporarily constituting an electric current. If charges are not replenished, the current ceases after a short time. Continuous current can occur if fresh charges are supplied, maintaining a steady electric field, typically provided by cells or batteries.
Ohm's Law
Ohm's Law, discovered by G.S. Ohm in 1828, relates current (I), potential difference (V), and resistance (R) in a conductor. It is expressed as V = R I, where R is the resistance measured in ohms (Ω). Resistance depends on the material and dimensions of the conductor.
Resistance and Its Dependence
Resistance (R) is proportional to the length (l) of a conductor. Doubling the length of a conductor doubles the resistance. - Relationship: R ∝ l Additionally, resistance is inversely proportional to the cross-sectional area (A) of the conductor. Halving the area of the cross-section doubles the resistance. - Relationship: R ∝ 1/A Combining these relationships gives: R ∝ l/A. For a given conductor, the resistance can be expressed as: R = ρ(l/A), where ρ is the resistivity, a constant that depends on the material but not on its dimensions.
Drift of Electrons and Resistivity
Electrons in a conductor experience collisions with fixed ions, resulting in random directions and an average velocity of zero. When an electric field (E) is applied, electrons are accelerated. The average velocity of electrons under the influence of an electric field can be derived from the velocities post-collision.
Charge Transport in Conductors
When an electric field E is applied, electrons experience a drift, leading to net charge transport across a given area A. The number of free electrons per unit volume is denoted as n. The total charge transported across area A in a time interval Δt is given by -neA|v_d|Δt. The current I is defined as I = |j|A, where |j| is the current density.
Current Density and Ohm's Law
The relationship between current density j and electric field E can be expressed as j = (ne/m)τE. This equation is a form of Ohm's law, where conductivity σ is defined as σ = (ne²τ/m). This indicates that electrical conduction can be described simply through the drift of electrons.
Mobility of Charge Carriers
Mobility (μ) is defined as the magnitude of the drift velocity per unit electric field: μ = |v_d| / |E|. Conductivity arises from mobile charge carriers, which include electrons in metals, and both electrons and positive ions in ionized gases and electrolytes.
Limitations of Ohm's Law
Ohm's law is valid for many materials, but there are exceptions where the proportionality of voltage (V) and current (I) does not hold. Deviations include: (a) V is not proportional to I. (b) The relationship between V and I depends on the sign of V, as seen in diodes. (c) There can be multiple values of V for the same I, as in Gallium Arsenide (GaAs). Despite these exceptions, many materials obey Ohm's law.
Resistivity of Various Materials
Materials are classified based on their resistivities: - Conductors: Low resistivities (10⁻⁸ Ωm to 10⁻⁶ Ωm). - Insulators: Very high resistivities (10¹⁸ times greater than metals or more). - Semiconductors: Intermediate resistivities that decrease with temperature. Doping semiconductors with impurities can further decrease their resistivity.
Temperature Dependence of Resistivity
Resistivity varies with temperature and is not uniform across materials. For metallic conductors, resistivity (ρ_T) can be approximated by the equation: ρ_T = ρ₀[1 + α(T - T₀)], where ρ₀ is the resistivity at a reference temperature T₀, and α is the temperature coefficient of resistivity. For metals, α is positive. The relationship is approximately linear over a limited temperature range, but deviates significantly at very low temperatures.
Electrical Energy and Power
In a conductor with endpoints A and B, where current I flows from A to B, the electric potential at A (V(A)) is greater than at B (V(B)). The potential difference (V) is defined as V = V(A) - V(B). During a time interval Δt, a charge ΔQ = I Δt moves from A to B, resulting in a change in potential energy (ΔU_pot).
Collisions and Energy Dissipation
In reality, charge carriers do not move with acceleration but with a steady drift velocity due to collisions with ions and atoms. During these collisions, energy gained by charges is shared with atoms, causing them to vibrate more vigorously and heating the conductor. The energy dissipated as heat in the conductor during time interval Dt is given by DW = I VDt.
Power Loss in Conductors
Using Ohm’s law (V = IR), the power loss (ohmic loss) in a conductor of resistance R carrying current I can be expressed as: P = I²R = V²/R. This power heats components like electric bulbs, causing them to radiate heat and light.
Cells and Steady Current Maintenance
A simple device to maintain a steady current in an electric circuit is the electrolytic cell, which consists of two electrodes (positive P and negative N) immersed in an electrolytic solution. The electrodes exchange charges with the electrolyte, providing the necessary energy for current flow.
Electromotive Force (emf)
The electromotive force (emf) of a cell is the potential difference between the positive and negative electrodes when no current is flowing. It is denoted by e and calculated as e = V+ + V–, where V+ > 0 and V– ≥ 0.
Current Flow and Internal Resistance
When a resistor R is connected across the cell, a current I flows from the positive electrode (P) to the negative electrode (N) through the resistor, while the current flows from N to P through the electrolyte. The electrolyte has a finite internal resistance r.
Combining Cells in Circuits
Cells can be combined in series or parallel in an electric circuit. Similar to resistors, a combination of cells can be replaced by an equivalent cell for calculating currents and voltages.
Kirchhoff’s Rules Overview
Kirchhoff’s rules are essential for analyzing electric circuits with multiple resistors and cells. They consist of two main rules: 1. **Junction Rule**: The sum of currents entering a junction equals the sum of currents leaving the junction. 2. **Loop Rule**: The algebraic sum of changes in potential around any closed loop is zero.
Wheatstone Bridge
The Wheatstone bridge consists of four resistors (R1, R2, R3, R4) and is used to measure unknown resistances. A battery is connected across one pair of diagonally opposite points (A and C), known as the battery arm. A galvanometer (G) is connected between the other two vertices (B and D), referred to as the galvanometer arm.
Balanced Bridge and Kirchhoff's Laws
In a balanced bridge, the condition I_g = 0 indicates no current through the galvanometer (G). Kirchhoff's junction rule gives I_1 = I_3 and I_2 = I_4.
Example Calculation
In Example 3.7, the resistances of the Wheatstone bridge are AB = 100Ω, BC = 10Ω, CD = 5Ω, and DA = 60Ω. A galvanometer with 15Ω resistance is connected across BD with a potential difference of 10V across AC. The equations derived from mesh analysis lead to the current through the galvanometer being calculated as I_g = 4.87 mA.