Title 16. Current Electricity.pdf ✅

Current Electricity 343 16 Current Electricity QUICK LOOK Electric Current: Current is a tensor quantity, while current density is a vector. Conventionally direction of current is taken along the direction of flow of positive charges. In metals charge carriers are only free electrons. In liquids charge carriers are positive and negative ions. In gases charge carries are positive ions and electrons. And in semi-conductors charge carriers are electrons and holes. Drift velocity of electrons in a metal is of the order of 3 10 / m s − and is directly proportional to electric field (or potential difference applied). The current flows with speed of light. Mean velocity of electrons due to their thermal agitations (or random motion) is zero; while mean speed depends on temperature. Figure: 16.1 ElectricCurrent q dq I t dt = = (scalar quantity) Current Density n I J A = where An = normal area Current . cos d I j A JA neAv = = = θ where d v is drift velocity. Ohm’s Law Under same physical conditions the voltage is directly proportional to electric current in dc circuits.V Rl = (Under same physical conditions) The resistance of a conductor is directly proportional to length and inversely proportional to cross-sectional area. i.e. At a given temperature, the specific resistance of a conductor is independent of dimensions but depends only on material. If a given mass of a material is stretched to decreases its cross-section, then its length also increase and then l R a α = or 2 l R r α Resistance 2 2 . l m l R A ne A ρ τ = = Where ρ = Specific resistance τ = Relaxation time, n = Electron density in meter–3 Stretching of Wire: If a conducting wire stretches, it’s length increases, area of cross-section decreases so resistance increases but volume remains constant. Suppose for a conducting wire before stretching it’s length 1 = l , area of cross-section 1 = A , radius 1 = r , diameter 1 = d , and resistance 1 1 1 l R A = ρ After stretching length 2 = l , area of cross-section 2 = A , radius 2 = r , diameter 2 = d and resistance 2 2 2 l R A = = ρ Ratio of resistances before and after stretching 2 2 4 4 1 1 2 1 2 2 2 2 2 1 2 1 1 1 R l A l A r d R l A l A r d         = × = = = =                 If length is given then 2 2 1 1 2 2 R l R l R l   ∝ ⇒ =     If radius is given then 4 1 2 4 2 1 1 R r R r R r   ∝ ⇒ =     Resistance of a conducting body is not unique but depends on it’s length and area of cross-section i.e., how the potential difference is applied. See the following figures Figure: 16.2 For length = a, area of cross-section = b × c Resistance a R b c ρ   =     × For length = b, Area of cross-section = a × c Resistance b R a c ρ   =     × Conductance 1 K R = b a c b c a d A Area Current measuring point vd e

Current Electricity 345 Figure: 16.6 Kirchoff’s second law (or voltage law) is based on conservation of energy. Loop law Σ = V 0 or Σ = Σ iR E for a closed circuit: Voltage Law: The net voltage. Drop around any closed looppath must be zero. Figure: 16.7 For any path you follow around the circuit, the sum of the voltages rises (like batteries) must equal the sum of the voltage drops. Voltage represents energy per unit charge, and conservation of energy demands that energy is neither created nor destroyed. Figure: 16.8 Equivalent Resistance for Cube: If a skeleton cube is made with 12 equal resistances each having resistance R then the net resistance across The longest diagonal ( AG or EC or BH or DF) 5 6 = R The diagonal of face (e.g. AC, ED... etc.) 3 4 = R A side (e.g. AB, BC etc.) 7 12 = R Figure: 16.9 Wheatstone’s Bridge: When Wheatstone’s bridge is balanced, the resistance in arm BD may be ignored while calculating the equivalent resistance of bridge between A and C. Condition of balance is P R Q S = Equivalent resistance between terminals connected to battery at balance 1 1 1 R P Q R S eq = + + + When battery and galvanometer arms of a Wheatstone’s bridge are interchanged, the balance position remains undisturbed while sensitivity of bridge charges. A Wheatstone’s bridge is most sensitive if its all resistance P, Q, R, S are equal. Metre Bridge: Unknown resistance 100 1 S R l − = × , where l = balancing length in cm Figure: 16.10 Potentiometer: If L is length of potentiometer wire AB, Potential gradient VAB k i L = = ρ , where ρ is resistance per unit length of potentiometer wire EMF of a cell, E kl = For same potential gradient 1 1 2 2 E l E l = Figure: 16.11 Combinations of Cells In series: If n identical cells are in series nE i R nr = + Where B Power supply A ε0 G R1 K1 Standard cell C G R R.B S A C P B Q E K 1 cm (100 – 1) cm i Q i2 i1 i = i1+ i2 P C R S G B i1 i2 D E H G E F D C A B R1 R2 V 0 VB V 0 0 V V R1 V V 0 R2 0 0 0 R1 R2 VB VB 0 I I I I I I I
346 Quick Revision NCERT-PHYSICS R = external resistance; r = internal resistance of a cell and E = emf of a cell Figure: 16.12 In parallel: n cells in parallel E i r R n =   +     Figure: 16.13 Mixed grouping: n cells in a now, m such rows in parallel mnE i mR nr = + Figure: 16.14 For maximum current R R ext = int or nr R m = If two cells of different emfs are correctly connected in series 1 2 1 2 E E i R r r + = + + If two cells of different emfs are wrongly connected in series i.e., (positive terminals connected together) 1 2 1 2 E E i R r r − = + + Some Standard Results for Equivalent Resistance Case (i): 1 2 3 4 1 2 3 4 5 1 2 3 4 5 1 2 3 4 1 3 2 4 ( ) ( ) ( )( ) ( ) ( )( ) AB R R R R R R R R R R R R R R R R R R R R R R R + + + + + + = + + + + + + Figure: 16.15 Case (ii): 1 2 3 1 2 3 1 2 2 ( ) 2 AB R R R R R R R R R + + = + + Figure: 16.16 Case (iii): 1/ 2 2 1 2 1 2 3 1 2 1 1 ( ) ( ) 4 ( ) 2 2 R R R R R R R R AB = + + + + +     Figure: 16.17 Case (iv): 2 1 1 1 1 1 4 2 AB R R R R     = + +         Figure: 16.18 Transformation between Y or Star and delta connection Figure: 16.19 The transformation from ∆ -load to Y-load. To compute the impedance Ry at a terminal node of the Y circuit with impedances R R', " to adjacent node i ' " y R R R R∆ = Σ n the ∆ circuit by ' " y R R R R∆ = Σ where R∆ are all impedances in the ∆circuit. This yields the specific formulae N R R R N N R R R N N N tends to infinity B A R1 R1 R1 RAB R2 R2 R2 tends to infinity B A R1 R1 R1 RAB R3 R3 R3 R2 R2 R2 R1 R2 R1 A B R2 R1 R2 R5 R1 A B R2 R3 R4 R5 n I I V R E r E r E r n E r E r E r m n I V R E r E r I n I I V R E r E r E r