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Electric Current

Home > Study Notes > Electric Current
Published in: AIEEE | IIT JEE Mains
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Notes On Electric Current.

Akhilesh K / Lucknow

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  1. ELECTRIC CURRENT Electric current: The net amount of charge flowing through a cross section in unit time is defined as current. Mathematical Equation: If rate of flow of charge is independent of time, then the electric current is said to be steady then I = Q/t If rate of flow of charge varies with time, the current at any time i.e. instantaneous current is given by I = dQ/dt . If n be the number of conduction electrons crossing cross —section in time t then I = ne/t . Sl unit of current is ampere. One ampere is defined as If one coulomb of charge flows across any of its cross-section in one second. Note: Electric current has direction as well as magnitude but it is not a vector quantity. This is because currents do not add like vectors. Ohm's Law: The current which flows in conductor is proportional to the potential difference which causes 'its flow at constant temperature and pressure. Thus IR. where the constant R is the resistance of conductor. Unit of resistance is Ohm denoted as Q . Resistance: 1 Formula for resistance is R = p Here I is length of conductor ., A is area of cross-section of conductor and p is resistivity of conductor . Resistivity of conductor is independent of size and shape of conductor, it depends on material of conductor and temperature and pressure. Unit of resistivity is Ohm-m. Inverse of resistivity is conductivity denoted by o unit is (Ohm-m)-l Current density: From equation V = IR - pJ1 Here J is current density defined as current per unit area (taken normal to the current) J = I/A. Sl unit of current density is A/m2 If there is angle between direction of current and area vector of cross section is 0 Then current density J = I / Acose
  2. Relation between Current density and Electric field . For formula V = IR — pJl But E = 1 E=pJ or J = OE Origin of resistivitv: Iln metallic conductors, the electrons in the outer shells are less bounded with the nucleus. Due to thermal energy at room temperature, such valence electrons are liberated from atom leaving behind positively charged ions. These ions are arranged in a regular geometric arrangement on lattice points. These liberated electrons collide with the ions. Or constantly gets scattered from its path causing resistivity of metallic conductors. Drift of electrons: In absence of electric field electrons move in randomly after colliding with ions and direction and velocities of electrons after collisions is such that sum of velocities is zero and net charge passing through any cross section is zero causing no electric current. In presence of electric field (E) electrons experiences electric force of magnitude Ee in the direction opposite to the direction of electric field thus the acceleration of electron is opposite to direction of electric field. Now F = ma thus acceleration of electron a = Ee/m this acceleration is momentary and becomes zero after collision, since electrons are continuously colliding with ions. And electrons get accelerated again and process goes of repeating. As a result electrons are dragged in the opposite to the electric field. Now average time period between two successive collisions is known as relaxation time t. And corresponding average velocity of electrons is known as drift velocity Vd. Now va = at Relation between drift velocity and current density Let us consider a cylindrical conductor of cross section A . Let E be the electric filed exists in conductor. If Vd is drift velocity of electrons then volume of the electrons passing through a cross- section in one second = VdA . If n is the number of electrons per unit volume then nvdA is the number of electron in passing through a cross-section in one second. Net charge passing through a cros-section in one second = neAvd = I . Now I = neAvd. Thus J = new
  3. Relation between resistivity and relaxation time. We know that J = o E, here o is conductivity And J =nevd nevd = o E Substituting value of Vd —T in above equation we get 2 ne m Since o=l/p ne2T Here we have assumed that and n are constant. On increasing temperature of conductor n does not change appreciable. The oscillations of ions increases with temperature and becomes more erratic. As a result, the relaxation time (T) decreases and resistivity of conductor increases with increase in temperature Mobility In case of conductors free electrons are mobile charge carriers. In case of electrolyte positive and negative ions are mobile charge carriers. In case of semiconductors holes and electrons are mobile charge carriers. Mobility is defined as drift velocity of charge per unit electric field IVdl S.l. unit of mobility is m2/Vs and is 104 of mobility in practical units (cm2 /Vs) —T in above equation we get By substituting va Limitations of Ohm's Law (1) V-l relations are not linear Example: diode transistor (2) The relation between V and I depends on the sign of V. If we change the polarity of supply voltage magnitude of current changes (3) The relation between V and I is not unique. There may be more values of potential for same current l. example tunnel diode, material Ga,As
  4. Classification of materials based on resistivity . Materials are classified as conductors, semiconductors and insulators depending on their resistivity. Conductors have resistivity of order 10-80m to 10-60m Insulator have resistivity 1018 times greater than metals or more Resistivity of semiconductors decreases with increase in temperature because covalent bond between adjacent atoms breaks which creates free electrons and holes causing decrease in resistivity. Or its conductivity increases. Commercially produced resistors for domestic use or in laboratory are of two major types: wire bound resistors and carbon resistors Wire bound resistors are made by winding the wires of an alloy viz. manganin, constantan, nichrome or similar ones. These alloys are relatively insensitive to temperature. These resistances are typically in the range of a fraction of an ohm or to a few hundred ohms Resistors, in the higher range are made mostly from carbon. Carbon resistors are compact, inexpensive and thus find extensive use in electronic circuit. Carbon resistors are small in size and hence their values are given using colour code Temperature dependence of resistivity: The resistivity of materials is found to be dependent on temperature. Over a limited range of temperatures, that is not too large, the resistivity of metallic conductor is approximately given by = poll + - To)] Where is the resistivity at temperature T and po is resistivity at temperature To, is called the temperature co-efficient of resistivity, dimension of a is [temperaturel-l units are (oc) 1 or (K) 1 Note that temperature coefficient of Carbon and, semiconductors germanium and silicon are negative, indicating with increase in temperature resistivity decreases. Equation of resistivity shows a linear relation between temperature and resistivity however graph of resistivity — temperature is not linear for copper. Since alloys like manganin, constantan, nichrome are relatively insensitive to temperature there graph is straight line intercepting on Y axis Resistivity of semiconductors decreases with temperature there graph is non linear and have negative slope indicating resistivity decreases with increase in temperature Resistivity Temperature T Copper Resistivi ty Temperature T nichrome Resistivity Temperature T semiconductor
  5. Cells emf and internal resistance Cell is a simple device which maintains a steady current in an electrical circuit. Cell consists of electrolyte and electrodes. Electrodes are N metallic plate dipped in electrolyte. Due to exchange of electrodes between electrolyte and electrode, one electrode develop positive potential deference (V+) between electrolyte and electrode and such electrode acts as positive terminal (P). While other electrode develop negative potential difference (V.) and acts as negative terminal (N). When there is no current, the electrolyte has the same potential throughout, so that the potential difference between P and N is -(-V-) = V++V- This difference is called electromotive force (emf) of the cell and is denoted by E. Thus emf is work done by non-electrical force in moving a positive charge from negative terminal to positive terminal of battery. Working of cell: Consider a wire of resistance R connected across the two terminals of the battery as shown in figure. The electric field is established in wire. As a result positive charge will move from higher potential (P) to lower potential (N) through external resistance R. The energy of the positive charge is consumed to overcome the resistance of wire. As it reaches the negative terminal N, its energy becomes zero, it is as per law of conservation of energy. Now due to non-electrical force positive charge on the negative electrode is moved towards the positive electrode inside the cell. Thus work is done by the non-electrical force and potential energy of positive charge increases when it reaches positive electrode. Again it flows through the external resistance R and process goes on repeating Now if external resistance is not connected then positive charge gets accumulated on the positive electrode and produces an electric field in the direction from positive electrode to negative electrode so that direction of electric force is opposite 'to non-electric force. When force due to electric field becomes equal to non-electric force flow of positive charge from negative terminal to positive terminal stops and potential across terminal is E Now during the discharge of cell positive charge has to overcome the resistance of electrolyte, such resistance is called internal resistance denoted by r. Terminal voltage When positive charge flows in electrolyte they have to overcome internal resistance t. if I is the current then energy lost in electrolyte is Ir. As a result voltage across P and N is less than E in the open circuit condition. The net energy per unit charge will be (E — Ir) . Thus , during the flow of current potential across between two terminal P and N is V = E — Ir. This potential difference is called terminal voltage. If E >> Ir , internal resistance is neglected We also observe that since V is the potential difference across resistance R, from Ohm's law V = IR Thus IR = E — Ir
  6. Maximum current can be drown from cell is R = 0. However, in most of the cells maximum allowed current is much lower to prevent permanent damage to cell. Internal resistance of electrolyte cell is very small. Thus electrolyte cell gives large value of current. Internal resistance of dry cell is higher thus it gives low current. Electrical Energy, Power Consider a conductor with endpoints A and B, in which a current I is flowing from A and B. The electric potential at A and B are denoted V(A) and V(B) respectively. Since current is flowing from A to B and potential difference across AB is In a time 'interval At, an amount of charge AQ = I At travels from A to B. The potential energy of charge at A = AQV(A) and The potential energy of charge at B = AQV(B) Thus change in potential energy AU = Final potential energy - Initial potential energy AU QV(B) - AQV(A) = - V(A)] = -AQV= -IV At O But when charges flow through the conductor they move with the steady drift velocity, because of collision with ions and atoms during transit. During collisions, energy gained by the charges thus is shared with the atoms. The atoms vibrate more vigorously, as a result conductor heats up. The amount of energy dissipated as heat in the conductor during the time interval is IV At The energy dissipated per unit time is the power P = AW/ At and we have p = IV Using Ohm's law V IR P = PR = v2/R Power loss is also called as "Ohmic loss"
  7. Power loss in transmission lines Consider a device of resistance R, to which power is to be delivered via transmission station. Power of device is P = VI or I = P/V Let resistance of transmission cable is Rc. The power dissipated in the connecting wires, which is wasted is Pc = 12Rc (02 Transmission cables from power stations are hundreds of miles and there resistance Rc is considerable. To reduce Pc voltage V is increased to very large value. Using such high voltage is dangerous, thus at user end voltage is reduced this can be achieved by transformers. Series Combination of resistors Two resistors are said to be connected in series, if only one end of their ends points are joined . In series connection current flowing through each resistor remains same, while sum of potential drop across resistors is equal to potential drop across combination 1 1 1 2 1 Consider three resistor RIR2 and R3 are connected in series as shown in figure. Potential difference across RI is IRI Potential difference across R2 is V2 = IR2 Potential difference across R3 is IR3 The potential difference across combination is V = VI + V2 + V3 V = I (RI + R2 + R3) If Req is equivalent resistance and V is potential difference across combination then I (Reg) Thus R eq = RI + R2 + R3 IF n resistance are connected in series the Req Parallel combination of resistors RI R2 12 13 I Two or more resistors are said to be connected in parallel if one end of all the resistors is joined together and similarly the other ends are joined together as shown in figure. In parallel combination current gets divided depending up on value of resistor, sum of current passing through the resistor is equal to current passing through combination, but potential difference across each resistor is equal to potential difference across combination Here Il, 12 and 13 are the currents passing through the resistors as shown in figure
  8. If I is the current passing through the combination then I 11+ 12 + 13 By applying Ohms law we get voltage across RI as V = 'IRI or Il = V/RI Similarly 12 = V/R2 and 13 = V/R3 By substituting values of current in above equation we get If combination is replaced by equivalent resistance Req , we would have from Ohm's law I = v/Req From above equations we have 1 Req 1 1 1 IF n resistors are connected in parallel the equivalent resistance 1 Req - Cells in series 1 1 1 1 Consider first two cells in series, where one terminal of the el two cells is joined together leaving the other terminal in either cell free. Let El and be the emf of the two cell having internal resistance rl and r2 respectively Let V (A), V (B), V (C) be the potentials at points A, B and C shown in figure. Then V (A) — V (B) is the potential difference between the positive and negative terminals of the first cell. We know that terminal voltage of first cell VAB = V(A) - V(B) = El - I rl Similarly for second cell VBC = V(B) - V(C) = €2 -l r2 Hence, the potential difference between the terminals A and C of the combination is VAc= v(A) - V(C) - v(B) + [V(B)- V(C) = El -l rl + (€2 -l r2) VAC =( El + €2) — I(rl + h) If we wish to replace the combination by a single cell between A and C of emf Eeq and internal resistance req , we would have VAC = Eeq — I req Comparing the last two equations, we get Eeq = El + and req = rl + r2 If polarity of second cell is reverse then Eeq = El - E2 (El > E2) Important notes-
  9. (i)The equivalent emf of a series combination of n cells is just the sum of their 'individual emf's, and (ii) The equivalent internal resistance of a series combination of n cells is just the sum of their internal resistances If n cells of emf E having internal resistance r each connected in series. Total current I if resistance R connected across combination Cells in parallel 81 Il 1 2 1 c 1 Consider a parallel combination of the cells as shown in figure h and 12 are the currents leaving the positive electrodes of the cells. At the point Bl, Il and 12 flow out whereas the current I flows in B2. Since as much charge flows in as out, we have I - -11+12 Let V (BI) and V (B2) be the potentials at BI and B2, respectively. Then, considering the first cell, the potential difference across its terminals is V (Bl) —V (B2). Hence V: v(B1)- El - El — V Points Bl and B2 are connected exactly similarly to the first cell. Hence considering the second cell, we also have V- v(B1)- E2- I 12 _ V Combining the last three equations 'we get El -v % -V rl r 12) 1 rl Erl) If we want to replace the combination by a single cell, between Bl and B2, of emf Eeq and internal resistance req, we would have V = Eeq - I req . Thus Eeq = eq In simpler way 1 eq Eeq eq 1 1 r2
  10. General formula for n cells 1 eq Eeq eq 1 1 1 If n cells of emf E having internal resistance r each connected in parallel. Total current I if resistance R connected across combination Mix grouping of cell me mR + r Let n identical cells be arranged in series and let m such rows be connected in parallel. Total numbers of cells are nm Emf of system = ne Internal resistance of the system = nr/m The current through the external resistance R Kirchhoff's rules mne mR + nr Electric circuits generally consist of a number of resistors and cells interconnected sometimes in a complicated way. 30 Q h Il 40 Q a 2 j C d 45 v 20 Q b IQ IQ Junction or Branch point: The point in network at which more than two conductors meet is called a junction or a branch point . Point a and d shown in figure are junction point . Loop: A closed circuit formed by conductors is known as loop. As shown in figure ihjdcbai forms a closed loop 80 v Kirchhoff's first rule or iunction rule: At any junction, the sum of the currents entering the junction is equal to the sum of currents leaving the junction. The proof of this rule follows from the fact that when currents are steady, there is no accumulation of charges at any junction or at any point in a line. Thus, the total current flowing in, (which is the rate at which charge flows into the junction), must equal the total current flowing out.
  11. Proof : Let QI, Q2 ...Q5 be electrical charges flowing through the cross-sectional area of the 12 Il 15 13 14 respective conductors in time interval t which constitute current Il, 12, 15 Hence QI = lit, Q = 12t, Q5 = 15t It is evident from figure that the total electric charge entering the junction is QI + Q , while Q2+ Q4 + Q5 amount of charge is leaving the junction in the same interval of time Thus QI + Q3 = Q2+ Q4 + Q5 . ht+ 13t = + + 15 t h +13-12- 14 - 15=0 Kirchhoff's second rule or Loop rule: The algebraic sum of changes in potential around any closed loop involving resistors and cells in the loop is zero Using law of conservation of energy and the concept of electric potential any closed circuit can be analyzed. In electric circuit, the electric potential at any point is a steady circuit does not change with time. Following are the sign convention has to be followed: (ii) If our selected path is in 'the direction of current then potential drop across resistor should be taken as negative else it should be taken as positive The emf of a battery should be considered negative while moving from negative terminal of a battery to the positive terminal. The emf of battery is taken as positive while moving from positive terminal while moving from positive to negative terminal of battery While analyzing the circuit we may get negative value of current indicates direction of current which arbitrarily chosen is opposite to the actual direction of current Wheatstone bridge 14 2 c RI The circuit shown in figure which is called the Wheatstone bridge. The bridge has four resistors RI, R2, R3 and R4 . Across one pair of diagonally opposite points (A and C in the figure) a source is connected. This (i.e., AC) is called the battery arm. Between the other two vertices, B and D, a galvanometer G (which is a device to detect currents) is connected. This line, shown as BD in the figure, is called the galvanometer arm. For simplicity, we assume that the cell has no internal resistance. In general there will be currents flowing across all the resistors as well as a current lg through G. Of special interest, is the case of a balanced bridge where the resistors
  12. are such that lg = O. We can easily get the balance condition, such that there is no current through G. In this case, the Kirchhoff's junction rule applied to junctions D and B (see the figure) immediately gives us the relations Il = 13 and 12 = 14. Next, we apply Kirchhoff's loop rule to closed loops ADBA and CBDC. The first loop gives ADBA —11 RI + 0 + o (Ig = o) and the second loop CBDC. 12R4 + O — llR3 = O 11 R2 12 - RI And using Il = 13 and 12 = 14 11 12 Hence at balanced condition from above equation This last equation relating the four resistors is called the balance condition for the galvanometer to give zero or null deflection. The Wheatstone bridge and its balance condition provide a practical method for determination of an unknown resistance. Let us suppose we have an unknown resistance, which we insert in the fourth arm; R4 is thus not known. Keeping known resistances RI and R2 in the first and second arm of the bridge, we go on varying R3 till the galvanometer shows a null deflection. The bridge then is balanced, and from the balance condition the value of the unknown resistance R4 is given by,
  13. Meter bridge The meter bridge is shown in Figure s 100 - Il Metre scale c It consists of a wire of length 1m and of uniform cross sectional area stretched taut and clamped between two thick metallic strips bent at right angles, as shown. The metallic strip has two gaps across which resistors can be connected. The end points where the wire is clamped are connected to a cell through a key. One end of a galvanometer is connected to the metallic strip midway between the two gaps. The other end of the galvanometer is connected to a 'jockeV. The jockey is essentially a metallic rod whose one end has a knife-edge which can slide over the wire to make electrical connection. R is an unknown resistance whose value we want to determine. It is connected across one of the gaps. Across the other gap, we connect a standard known resistance S. The jockey is connected to some point D on the wire, If the jockey is moved along the wire, then there will be one position where the galvanometer will show no current. Let the distance of the jockey from the end A at the balance point be ll. The four resistances of the bridge at the balance point then are R, S, p h and p (100— h). The balance condition, R pli 11 S - p(100 - 11) 100 - 11 Thus, once we have found out Il, the unknown resistance R is known in terms of the standard known resistance S by 11 100 - 11 By choosing various values of S, we would get various values of Il, and calculate R each time. An error in measurement of Il would naturally result in an error in R. It can be shown that the percentage error in R can be minimized by adjusting the balance point near the middle of the bridge, i.e., when Il is close to 50 cm.
  14. Potentiometer Construction It is basically a long piece of uniform wire, sometimes a few meters in length across which a standard cell is connected. In actual design, the wire is sometimes cut in several pieces placed side by side and connected at the ends by thick metal strip. In the figure, the wires run from A to C. The small vertical portions are the thick metal strips connecting the various sections of the wire. A current I flows through the wire which can be varied by a variable resistance (rheostat, R) in the circuit. Since the wire is uniform, the potential difference between A and any point at a distance I from A is E I (where is the potential drop per unit length. Figure shows an application of the potentiometer to compare the emf of two cells of emf El and E2 . The points marked 1, 2, 3 form a two way key. Consider first a position of the key where 1 and 3 are connected so that the galvanometer is connected to El. The jockey is moved along the wire till at a point NI, at a distance h from A, there is no deflection in the galvanometer. We can apply Kirchhoffs loop rule to the closed loop ANIG31A and get, Use to compare emf 1 c Il + O— El = O Similarly, if another emf E2 is balanced against 12 (AN2) 12 + O— E2 = O From the last two equations 11 El — 12 Thus we can compare the emfs of any two sources. In practice one of the cells is chosen as a standard cell whose emf is known to a high degree of accuracy. The emf of the other cell is then easily calculated from above equation
  15. use to measure internal resistance of a cell B. c For this the cell (emf E ) whose internal resistance (r) is to be determined is connected across a resistance box through a key 1

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