Tiêu đề Current Electricity.pdf ✅

CURRENT ELECTRICITY Introduction Charges in motion constitute an electric current. In the present chapter, we shall study some of the basic laws concerning steady electric currents and will analyse some electrical circuits. Electric Current The rate of flow of electric charge across any cross-section is called electric current. (a) Instantaneous electric current I = dq dt (b) Average electric current Iav = Δq Δt (c) SI unit of current is ampere (A) and CGS-emu unit is biot (Bi), or ab ampere. 10 ampere = 1 biot (d) The conventional direction of current is taken to be the direction of flow of positive charge and is opposite to the direction of flow of negative charge. (e) Although electric current is denoted with direction in dc circuits but it is a scalar quantity. Direction merely represents the sense of charge flow. (f) Current carriers: The charged particles whose flow in a definite direction constitutes the electric current are called current carriers. In different situation current carriers are different. (i) Solid conductors: In solid conductors like metals current carriers are free electrons. (ii) Electrolytes or liquid conductors: Current carriers are positive and negative ions. (iii) Gas discharge tube: Current carriers are positive ions and free electrons. (iv) Semi conductor: Current carriers are holes and free electrons. (g) When current ( I ) at some instant is known we can obtain the charge that would pass through a cross - section of a wire between time interval t1 to t2 as q = ∫ t2 t1 Idt For steady current q = I(t2 − t1 ) For example, if the current in a wire varies with time according to the equation I = 4 + 2t, where I is in ampere and t is in sec. Then the total charge passed between interval t = 2 s and t = 6 s is q = ∫ 6 2 (4 + 2t)dt = 48 coulomb Current Density In a current carrying conductor we can define a vector which gives the direction as current per unit normal, cross sectional area. Thus, J ⃗ = I A nˆ Where nˆ is the unit vector in the direction of the flow of current.
Current though a wire can be obtained as I = J ⃗ ⋅ A⃗⃗ = JAcos θ (where θ is the angle between J ⃗ and A⃗⃗ ) For variable J, current I = ∫ J ⃗ ⋅ dA⃗⃗ Drift Velocity Average velocity with which electrons drift from low potential end to high potential end of the conductor ( vd ). Drift velocity is given by v⃗d = − eτ m E⃗⃗ (in terms of applied electric field) where e is magnitude of electron's charge, m is the mass of electron and τ is the average time between two successive collisions called as mean relaxation time. Above expression of drift velocity tells us that the electrons move with an average velocity which is independent of time, although electrons are accelerated. If vd be the drift velocity and ' n ' be the number of such free electrons per unit volume, the current through the conductor of cross sectional area A is I = neAvd Ohm's Law It states that current flowing between two point in a conductor is directly proportional to the potential difference between the two points. I ∝ V, provided temperature is constant ⇒ V I = constant (R) ⇒ V = IR The constant ' R ' is called resistance of the conductor. Vector form of ohm's law is stated as Current Density ∝ Electric Field i.e. J ⃗ ∝ E⃗⃗ or J ⃗ = σE⃗⃗ Where σ is a constant of proportionality called conductivity (unit Ω −1 m−1 or mho/m or siemen/m). Electrical Resistance For a conductor of uniform cross-sectional area A, resistance between two ends separated by length l is given by RAB = ρ l A where, ρ = Resistivity of the conductor l = length of the conductor in the direction of current flow A = Uniform area of cross-section perpendicular to current flow Units: R → ohm(Ω), ρ → ohm − meter(Ω − m) also called siemens. (1) Series Grouping
In this type of grouping current remains undivided through each resistance but potential difference distributes in the ratio of resistance i.e., V ∝ R Using Kirchhoff's loop rule for the loop shown in figure, V = IR1 + IR2 + IR3 Let the equivalent resistance between of the circuit is equals Req , by definition. V = IReq Hence equivalent resistance is obtained as Req = R1 + R2 + R3 (i) Equivalent resistance is greater than the maximum value of resistance in the series combination. (ii) If n identical resistance are connected in series Req = nR and potential difference across each resistance V ′ = V n (iii) In series combination voltage across each resistor can be obtained as (2) Parallel Grouping
In this type of grouping potential difference remains undivided across each resistance but current distributes in the reverse ratio of their resistance i.e. , I ∝ 1 R Applying Kirchhoff's junction law at point P Or I = I1 + I2 + I3 Req = V R1 + V R2 + V R3 Hence, equivalent resistance is given by 1 Req = 1 R1 + 1 R2 + 1 R3 or Req = R1R2R3 R1R2 + R2R3 + R2R1 (i) Equivalent resistance is smaller than the minimum value of resistance in the parallel combination. (ii) If two resistance in parallel Req = R1R2 R1 + R2 = Multiplication Addition (iii) Current through any resistance I ′ = I × [ Resistance of opposite branch Total resistance ] Where I ′ = required current (branch current), I = main current I1 = I ( R2 R1 + R2 ) and I2 = I ( R1 R1 + R2 ) (iv) In n identical resistance are connected in parallel Req = R n and current through each resistance I ′ = I n Emf of cell ( E ): The potential difference across the terminals of a cell when it is not supplying any current is called it's emf. Terminal voltage ( V ): The voltage across the terminals of a cell when it is supplying current to external resistance is called potential difference or terminal voltage. Internal resistance ( r ): In case of a cell the opposition of electrolyte to the flow of current through it is called internal resistance of the cell. Kirchhoff's Laws (1) Kirchhoff's First Law