Complete notes for Chapter 13. Covers Electric Current, Ohm's Law, Resistivity, Kirchhoff's Laws, Wheatstone Bridge, Potentiometer, and Electrical Power.
Electric Current: Time rate of flow of charge through any cross section of conductor is called electric current.
Mathematically: If ΔQ charge passes through any cross section of a conductor in time Δt, then the electric current I is given by:
$$ I = \frac{\Delta Q}{\Delta t} $$
Electric current is a scalar quantity. Although we assign it a direction, it does not obey the law of vector addition.
The SI unit of current is Ampere (A). When one coulomb charge passes through any cross section of a conductor in one second, the current is said to be one ampere.
$$ 1A = \frac{1C}{1s} $$
The conventional current in a circuit is defined as that current which passes from a point at higher potential to a point at lower potential as if it represented by a movement of positive charges.
The electronic current in a circuit is that current which passes from a point of low potential to a point of high potential as if it represented by a movement of negative charges.
Note: Experimentally, positive charges moving in one direction are equivalent in all external effects to negative charges moving in the opposite direction.
The valence electrons in metals are not bound to individual atoms but are free to move. These are called free electrons. Their random motion depends on temperature.
The average constant velocity with which free electrons drift in a conductor under the influence of an electric field is called drift velocity.
It is of the order of \( 10^{-3} m/s \) or \( 1 mm/s \).
Why constant velocity? The force does not produce net acceleration because electrons collide with lattice atoms, transferring energy (heat) to them.
A source of current maintains a constant potential difference across a conductor. It converts non-electrical energy into electrical energy.
Current flows due to electron motion. Collisions with atoms transfer kinetic energy to the lattice, increasing vibrational energy and producing heat.
Joule's Law: $$ H = I^2Rt $$
Applications: Electric heaters, toasters, electric irons.
Passing current through a wire produces a magnetic field around it. Strength depends on current magnitude and distance.
Applications: Motors, measuring instruments.
Conduction through liquids (electrolytes) via chemical reactions (electrolysis).
Electrolysis Components: Electrolyte (liquid), Electrodes (Anode/Cathode), Voltameter (vessel).
Electroplating: Process of coating a thin layer of expensive metal (Gold, Silver) on a cheap metal (Iron) using proper electrolyte and electrodes (Anode = coating metal, Cathode = object).
The current flowing through a conductor is directly proportional to the potential difference across its ends, provided the physical state (such as temperature) of the conductor does not change.
$$ V \propto I $$
$$ V = IR $$
Where R is the constant of proportionality called Resistance.
The opposition to the flow of charge through a conductor is called electrical resistance. Its SI unit is Ohm (Ω).
Definition of Ohm: Resistance is said to be one ohm if a current of one ampere flows through a conductor when a potential difference of one volt is applied across its ends.
$$ 1 \Omega = \frac{1V}{1A} $$
A conductor which obeys Ohm's law strictly is called an Ohmic conductor. The graph between V and I is a straight line passing through the origin. Examples: Metallic conductors (Copper, Silver, Gold).
A conductor which does not obey Ohm's law is called a Non-Ohmic conductor. The V-I graph is not a straight line.
Resistors are connected end-to-end. Same current passes through all.
$$ R_{eq} = R_1 + R_2 + R_3 + ... $$
Resistors are connected side-by-side with ends joined common. Potential difference across each is same.
$$ \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + ... $$
Resistance is directly proportional to length (L) and inversely proportional to area (A). $$ R = \rho \frac{L}{A} $$
Where ρ (rho) is Resistivity. It is the resistance of a meter cube of a material.
Unit: Ohm-meter ($$ \Omega m $$).
Dependence: Depends on nature of material and temperature. Independent of dimensions.
Reciprocal of resistance. $$ G = \frac{1}{R} $$
Unit: Mho ($$ \Omega^{-1} $$) or Siemens (S).
Reciprocal of resistivity. $$ \sigma = \frac{1}{\rho} $$
Unit: $$ (\Omega m)^{-1} $$ or $$ S m^{-1} $$.
Effect of Temperature: As temperature rises, amplitude of atomic vibrations increases, increasing collision probability with free electrons. Hence, resistance increases.
Temperature Coefficient (alpha): The fractional change in resistance per kelvin.
$$ \alpha = \frac{R_t - R_o}{R_o t} $$
Where $$ R_o $$ is resistance at $$ 0^\circ C $$, $$ R_t $$ is resistance at $$ t^\circ C $$, and $$ t $$ is temperature change.
Resistivity Coeff: $$ \alpha = \frac{\rho_t - \rho_o}{\rho_o t} $$
Unit: $$ K^{-1} $$
Carbon resistors use colour bands to indicate value. Code: BBROYGBVGW (Black 0, Brown 1, Red 2, Orange 3, Yellow 4, Green 5, Blue 6, Violet 7, Grey 8, White 9).
A wire-wound variable resistor. Consists of manganin wire on an insulating cylinder.
Heat-sensitive resistor. detailed construction from semiconductor ceramics (Ni, Mn, Co, Fe oxides).
Temperature sensors (e.g., 10K detection), voltage stabilization, electronic circuits.
The rate at which work is done to maintain steady current.
$$ P = VI $$
Using Ohm's law:
$$ P = I^2R = \frac{V^2}{R} $$
Unit: Watt (W).
Energy supplied by the battery to a unit charge moving from negative to positive terminal (inside source).
$$ E = \frac{\Delta W}{\Delta q} $$
Cause: Converts non-electrical energy to electrical.
Energy dissipated per unit charge across external resistance.
Relation: $$ V_t = E - Ir $$
Where r is internal resistance. When switch is open (I=0), $$ V_t = E $$.
Power delivered to load R is max when Internal Resistance (r) equals Load Resistance (R).
$$ P_{max} = \frac{E^2}{4r} $$
Sum of all currents meeting at a point in a circuit is zero.
$$ \sum I = 0 $$
Conservation: Law of Conservation of Charge.
Convention: Flowing towards a point = Positive. Flowing away = Negative.
The algebraic sum of potential changes along a closed loop is zero.
$$ \sum V = 0 $$
Conservation: Law of Conservation of Energy.
Rules:
A circuit to measure unknown resistance.
Four resistors R1, R2, R3, R4 forming a loop. A galvanometer connected between B and D.
When the bridge is balanced (no current in galvanometer, Ig=0), the potential at B equals potential at D.
$$ \frac{R_1}{R_2} = \frac{R_3}{R_4} $$
If R4 is unknown (Rx), then:
$$ R_x = R_3 \frac{R_2}{R_1} $$
A device to measure or compare potentials without drawing any current from the circuit (acts as ideal voltmeter).
A wire of length L and resistance R with a sliding contact C.