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Electric Current The rate of flow of charge through a cross-section of some region of a metallic wire (or an electrolyte) is called the current through that region . Current at any instant is given bv I - daldt. If a charge q flows through the circuit for time t, then SI unit of current is ampere (A) or (Cs-1) In metallic conductors, the current is due to the motion of free electrons whereas in electrolytes and ionized gases, both positive and negative ions are responsible for current flow.
Drift Velocity . When an electric field is applied across a conductor, the free electrons are accelerated opposite to the direction of the field and therefore they have a net drift in that direction. Due to frequent collisions of electrons with the atoms, their average velocity is very small. This average velocity with which the electrons move in a conductor under an applied potential difference is called the drift velocitv, The acceleration of the electron is a e E/m
o The drift velocity must be E - applied electric field m-mass of electron relaxation time . Current: I neAva n Number density of electrons .Mobility : The drift velocity per unit electric field.
Current Density Current density at a point is defined as the amount of current following per unit area around that point, provided the area is normal to the direction of current. J-I/A, where I is current and A is area.
Current is perpendicular to area: The average current density is given by Ja,- . Current not perpendicular to area: The current density is given by av AI average av AAcos0
Ohm's Law .The current flowing through a conductor is proportional to the potential difference across its ends, only when physical conditions (such as temperature) remain unchanged, i.e., Voci or V = RI The constant of proportionality R is called the resistance of the conductor
o Resistance: The resistance of a conductor is a measure of the opposition offered bv the conductor to the flow of current. . Theresistance of a conductor is directly proportional to its length / and inversely proportional to the area of cross-section A, i.e., where the constant depends on the nature of the material, is called the resistivity. In vector form I V El E where = is called conductivity.
Resistivity: The resistivity of material of a conductor is given by In 2 ne T where n is number of free electrons per unit volume and is the relaxation time of the free electron.
Temperature Dependence of Resistivity If po and pr are the values of resistivity at 0 C and T C respectively then over a temperature range that is not too large, we have approximately, Consequently, for resistance we have, approximately RT= R0(1 + ).
Colour Code of Resistors Colour Figure black brown red orange yellow green blue violet grey white 0 132 decimal multiplier (10") first two significant figures (SS2) tolerence of resistance (e) 4 Colour gold 8silver Tolerence 5% 10% no colour | 20%
Combination of Resistance Combination of Resistance Series combination Parallel combination Current is same across each resistance Potential drop across each reasistance is same. 1=11+ 12 + 13 Rp R R R Rp R R2 R3 > Rs = R1+ R2 + R3 Rmax(R,/R2/R) Rp < min(R,/R2/R3)
e Current divider : In a parallel circuit, potential drop across each resistance is same. 11R 2R2 On solving eqns. (i) and (ii), we get R, +R Note: Current through any resistance is Resistance of opposite branch Total resistance I = 1 total
Resistances in special cases Geometry| Object description Resistance of Object description Resistance of of object A solid cylinder object of object Rectangular slab object RABP A hollow cylinder Solid cylinder with different densities and 2 22 A cubical frame with equal resistance r on edges 12 12 RP2 4 Net R2 R +R2 17
EMF (Electro-Motive Force) . The potential difference across the terminals of a cell when it is not producing any current is called emf of the cell Emf depends on nature of electrolyte metal of electrodes Emf does not depend on area of plates distance between the electrodes quantity of eletrolyte ize of cell Terminal voltage : When current drawn through the cell or current is supplied to cell, the potential difference across its terminals is its terminal voltage.
Potential difference (V) across the terminals of ce Situation Discharging of a battery Charging of a battery Battery is open circuited | VAB- as 1-0 Battery is short circuited
Combination of Cells (Battery) Combination of Cells (Battery) Series Combination Parallel Combination Mixed Combination (Identical cells) e, ri n cells Equivalent emfeeq- + ,- Equivalent resistance, ( )/5) 1'2 2'1 Current in the circuit nr ir 12 eq e, +e, eq nr
KIRCHHOFF'S LAWS FOR ELECTRICAL NETWORK Junction Rule The algebraic sum of currents meeting at any junction in a circuit is zero, i.e., >I Conventionally the incoming currents at the point are taken as positive while those outgoing are taken as negative. In figure for junction O 0 4 1 =11 + 12-13 + 14-15-0 This law is based on conservation of charge.
Loop Rule According to this law in any closed part of arn electrical circuit, the algebraic sum of the emfs is equal to the algebraic sum of the products of the resistance and current flowing through them. This law follows from the law of conservation of energy.
Wheatstone's Bridge In a balanced condition even though current A flows in the rest of the circuit, galvanometer r wil not show any 4. W1 deflection (i.e. I,-0) Also, Ri R3 R2 R 4
Sol.: In steady state condition, no current flows through the capacitor C(5 F). Obviously no current flows through R1 (62) 6 6 V 2 11 4 1.5 A 3 V 3
Current through R3 /R3 Assume this I (1.5 A) divides as 1 and I2 as shown. Apply Kirchhoff's law to closed loop DGHFED 3-31,-(4 1.5)-211 + 2-0 or 511 =-1 or! The negative sign of 11 shows that its direction is reverse 6 6 V C 2 11 4 1.5A F of the direction shown in the figure. VA-211 + 2-VD = 0 along passage ABED 11 or VA-VD-211-2-2(-0.2)-2=-2.4 V H or VD- V 2.4 V 3 .. Potential difference across capacitor2.4 V Energy stored in capacitor =-CV or U-x (5 x 10-6) x (2.4)2 or U =- 5 10-o 5.76 = 1.44 10-5 J
Illustration 5 : Five equal resistances, each of resistance R, are connected as shown inR figure below. A battery of V volt is connected between A and B. Find the current flowing in FC. D ( -ww N
Sol.: The equivalent circuit diagram is as shown irn igure It is a balanced Wheatstone bridge, hence no current flows in arm CD. Therefore resistance ofarm CD becomes ineffective. I Resistan e6 p p r arm FCE =R+R=2R Resistance of lower arm FDE RR 2R The equivalent resistance between A ad B is 1/2 I/2 A F 1/2 e2R2R Cn rent in th circu t is 1 = .. Current in FC- 2 2R
Illustration6: A copper wire having cross-sectional area of 0.5mmand a length of 0.1 m is initially at 25 C and is thermally insulated from the surrounding If a current of 10 A is set up in this wire, (i) find the time in which the wire will start melting. The change of resistance with the temperature of the wire may be neglected. (ii) What will this time be, if the length of the wire is doubled ? Melting point of copper 1075 C. Specific resistance of copper 1.6 10-0 m Density of copper - 9 x 10 kg m Specific heat of copper-98 10-2 cal kg-8 oc-
Sol. : Mass of wire (m) = area length density -0.5 x 10-6 x0.1 x9 x 103-45 x 10-5 kg Rise in temperature ( ) 1075 C-25 C 1050 C (i) If a current I flows in a wire of resistance R for a time t, the heat energy generated is I Rt _ Ca 4.2 If s is the specific heat of copper, the wire will melt in time t given by I Rt The resistance R is given by
Sol.: We know that charge on the capacitor at any time is given by q = q0 (1-e-t/ ) where q0 C-15C. Here charge q at any time is given by q - VC where V is potential difference across capacitor at that time. Here V-5 V, so q-5C. Putting the values, we get -t/t 5C 15C(1 -oret- 2/3 -t/t RC 2 -6 Or R ln(3/2) 10 ln (3/2)
Devices and their principle Meter Bridge Principle It is based on Wheatstone bridge. Application (i) To measure unknown resistance, Circuit diagram x R100-D i To compare two unknown resistances |A RL = 12 (100-11) R2 (100-2) (iii) To measure the unknown temperature D100-I Metre scale x100 Potentiometer (i) Comparision of emf of two cells Principle Potential difference across any segments Length of the segment (ii) Internal resistance of given primary cell It is a safety device that protects the Fuse Principle Heat produced by appliance from getting damaged, by electric current in wire H- Rt Fuse melting and opening the circuit. appliance
Illustration 8: Cells A and B and a galvanometer G are connected to a slide wire OS by two sliding contacts C and D as shown in fi long and has a resistance of 12 S2. With OD 75 cm, the galvanometer gives no deflection when OC is 50 cm. If D is moved to touch the end of wire S, the value of OQC for which the galvanometer shows no deflection is 62.5 cm. The emf of cell B is 1.0 V. Calculate (i) the potential difference shown in figure. The slide wire is 100 cm across O and D when D is at 75 cm mark from O (ii) the potential difference across OS when D touches S (iii) internal resistance and emf of cell A