1 Chapter : 3 Current Electricity Current Electricity The branch of Physics which deals with the study of electric charges in motion is called current electricity. Electric current The flow of electric charges in a particular direction constitutes electric current. The electric current is defined as the rate of flow of charges across any cross sectional area of a conductor is known as electric current. Where, I = electric current q= charge passed through any cross section of conductor t= time If the rate of flow of charge is not uniform then current varies with time and is given by- Note Current is a scalar quantity. The direction of conventional current is taken as the direction of flow of positive charges or opposite to the direction of flow of electrons. Units of electric current: Ampere (in S.I. System)- One ampere represents the passage of one coulomb of charge per second. An electric current is due to the drift of - 1. Electrons in a metallic conductor. 2. Positive and negative charges in an electrolyte. 3. Electrons and ions in gases in discharge tubes. 4. Electrons and holes in a semiconductor.
2 How Electric Current Flow in the Conductor When no potential difference is applied across a conductor, the electrons are in random motion. The average velocity of electrons is zero. Thus the motion of electrons does not constitute any transport of charge in any direction. The current in the conductor is zero. When potential difference is maintained (i.e. battery is connected) across a conductor, the electrons gain some average velocity in the direction of positive potential (towards positive terminal of battery). And thus current is set up in the conductor. Ohm s Law It states that physical conditions like temperature pressure etc. remaining constant, the current flowing through a conductor is always directly proportional to the potential difference across the conductor. i.e. or I V V I or V = IR (1) Where R is constant of proportionality, called resistance of conductor. From (1) = constant So, if we plot a graph between V and I it will be a Straight line as shown in figure. Resistance It is the property of a body due to which it opposes the flow of current through the conductor. It is equal to ratio of potential difference applied to the current flowing through the conductor.i.e. Unit of R: unit of resistance is ohm (Ω) in S.I. system. One Ohm: Resistance of a conductor is said to be one ohm if a potential difference of one volt produces one ampere current through it. Cause of Resistance: As we know that there are large numbers of free electrons in a conductor. When a potential difference is applied across the ends of conductor the free electrons moves with higher energy and collide among themselves and with the atoms of conductor and causes obstacle
3 to flow of current. resistance. This obstacle produces the opposition to flow of electrons i.e. produces Conductance (G) The reciprocal of resistance is known as conductance. i.e. G = Unit of conductance is Simen or mho. Resistivity ( ) As we know, resistance of a conductor (R) is (i) Directly proportional to length(l) of conductor i.e. R l (ii) Inversely proportional to the area of cross-section (A) of the conductor R Combining these two points, we have R = Where, is the constant of proportionality, called resistivity of the material. If l = 1, A = 1 then = R So, is defined as the resistance of a material having unit length and unit area of crosssection. depends on the nature of material, but not on the dimensions of the material. Unit of is ohm-metre Conductivity ( ) The reciprocal of resistivity ( ) is called conductivity ( ) i.e. = Unit of conductivity is Simen per metre Current density (J) The electric current per unit area,taken normal to the current, is known as Current Density.
4 J = Unit of J is Ampere/metre 2 The current density is a vector quantity directed along the flow of current. When the plane of small area A makes an angle with the direction of current, then Or J = I = JA cos Or I = If E is the magnitude of uniform electric field in the conductor whose length is l, then the potential difference V across its ends is given by- Also V = I R = I Or E l = J l Or E = J In vector form V = E l (as Or Where is conductivity. Drift Velocity When no Electric field is applied across a conductor, the electrons are in random motion. The average velocity of electrons is zero. Thus the motion of electrons does not constitute any transport of charge in any direction. The current in the conductor is zero. But when Electric field is applied, the free electrons at negative end experience a force F = ee in a direction opposite to the electric field i.e. towards positive end. Thus due to collision, there act backward force on electron. So, the electron drift slowly towards positive end with a constant average velocity called, drift velocity v d So, drift velocity is defined as the velocity with which free electrons get drifted towards the positive terminal, when electric field is applied is known as drift velocity Expression for drift velocity: If = average time between two successive collisions a = acceleration produced Then, =
5 Also = (F= force on electron, m= mass of electron) But = - e where negative sign shows that direction of force is opposite to that of electric field. So, a = Or = where negative sign shows that direction of drift velocity is opposite to that of electric field. Which is expression for drift velocity. Note Mobility: Magnitude of drift velocity is given by v d = v d = E Or v d = where = is mobility, and is defined as the drift velocity acquired per unit electric field. Unit of mobility= m 2 V -1 s -1 Relation between current and drift velocity Consider a conductor of length l and area of cross section A. An electric field E is applied between its ends. Let n be the number of free electrons per unit volume. The free electrons move towards the left with a constant drift velocity v d. The number of conduction electrons in the conductor = nal The charge of an electron = e The total charge passing through the conductor q = (nal) e The time in which the charges pass through the conductor, t = Current flowing through the conductor, I = =
6 So, I = nae Or = ne Or J = ne This is the relation between current and drift velocity. Note: J = ne J = in vector form = Also = Comparing above two equations = Deduction of Ohm s Law: The current flowing through a conductor is given by, I = nae Also v d = Therefore, I = nae Or I = na V as Where = constant (= resistance) Therefore I V Which is ohm s law. Try example 3.1 and 3.2 from NCERT BOOK Download it from or website of ncert Limitations of Ohm s Law 1. For some good conductor graph between V and I is not exactly linear. 2. In diodes relation between V and I depends on sign of V. 3. For semiconductor graph between V and I is not linear.
7 Note: the conductors which do not obey ohm s law completely are called non-ohmic conductor e.g. semiconductors, and diodes. RESISTIVITY OF VARIOUS MATERIALS Depending on resistivity, the materials can be classified as conductor, semiconductors and insulators. The conductors (metals) have low resistivities in the range of 10-8 Ωm to 10-6 Ωm. The insulators have very high resistivities in the range of 10 8 Ωm to Ωm. The semiconductors have resistivities between conductors and insulator i.e. in the range of 10-2 Ωm to 10 4 Ωm. Temperature Dependence of Resistance As we know for metallic conductor R = i.e. R So, when temperature of a conductor is increased, the thermal energy of electron increases. Due to which the frequency of collision of free electrons with atoms or ions also increase and hence decreases. Hence, resistance increases. If R 0 = resistance of conductor at 0 0 C & R t = resistance of conductor at t 0 C And t = rise in temperatue. Then R t = R 0 ( 1 + t) ---(1) Where = temperature coefficient or resistance. From (1) R t = R 0 + R 0 t Or = i.e. Temperature coefficient of resistance = Hence is defined as the change in resistance per unit original resistance per degree rise in temperature.
8 Unit of = 0 C -1 or kelvin -1 For metals: is positive. So, from (1) R t > R 0 Therefore resistance of metals increases with rise of temperature.(this implies that conductivity decreases with rise in temperature.) For insulators and semiconductors: is negative. So, from (1) R t < R 0 Therefore resistance of insulators or semiconductors decrease with rise of temperature.(this implies that conductivity increases with rise in temperature.) For Alloys: for alloys like Nichrome, Manganin etc is very-very small. So, from (1) R t R 0 Therefore resistance of alloys almost remains same with rise in temperature. That is why we use these alloys to make standard resistance coil. Similarly resistivity of a metallic conductor is given by t = 0 ( 1 + t) Colour Code of Carbon Resistors The value of resistors and their percentage accuracy are indicated on carbon resistor by colour code printed on them. The colour code of carbon resistors is as given in the following table- Colour black brown red orange yellow green blue violet grey white gold silver No Mult Colour green silver No Colour Tolerance 5% 10% 20% Method1: when there is a set of rings of colours on the carbon resistors 1 st ring colour gives the first significant figure 2 st ring colour gives the second significant figure 3 rd ring colour gives the multiplier 4 th ring (if any) gives the tolerance
9 e.g. if 1 st, 2 nd,3 rd and 4 th colour are red, yellow, orange and golden respectively, then its value will be % Method2: Colour of body gives the first significant figure Colour of ends gives the second significant figure Colour of dots gives the multiplier Ring (if any) gives the tolerance e.g. if body colour be blue, end colour be green, dot be red and ring is silver, then its value will be % COMBINATION OF RESISTANCES Resistances in Series Resistance are said to be connected in series, if the same current is flowing through each resistor, when some potential difference is applied across the combination. Let, and are three resistances connected in series as shown in figure If, and is the potential drop across, and respectively. Then, V = (1) Also from Ohm s Law, V = I where is the total resistance of series combination. So,, (2) So, from (1) and (2), we have I = I + I + I Or = + + If there are n resistances connected in series then = Hence if a number of resistances are connected in series then their resultant resistance is equal to sum of individual resistances.
10 Resistances in Parallel A number of resistances are said to be connected in parallel if potential difference across each of them is the same and is equal to applied potential difference. Let, and are three resistances connected in parallel as shown in figure. Let V = potential difference applied. If, and is the current flowing through, and respectively. Then, Then total current, I = + + Also V = = = Or,, Or + + where is total resistance of parallel combination. Or + + Similarly, if a number of resistances are connected in parallel, then the resultant resistance is given by Hence, the reciprocal of resultant resistance in parallel is equal to sum of reciprocals of the individual resistances. Terminal Potential Difference The potential difference between two electrodes of the cell in closed circuit i.e. when current is being drawn from it, is called terminal potential difference. E.M.F. Electromotive force is the maximum potential difference between two electrodes of the cell in poen circuit i.e. when no current is being drawn from it. Internal Resistance of a Cell The internal resistance of a cell is the resistance offered by electrolyte and electrodes of a cell when electric current flows through it.
11 Internal resistance depends on- 1. Distance between the electrodes 2. Nature of electrolyte 3. Nature of material of electrodes 4. Area of electrodes immersed in electrolyte. Expression for Internal Resistance Consider a circuit with a battery, key and a resistance R as shown in figure. Let E = emf of the cell r = internal resistance of the cell. R = resistance connected & I = current clowing through the circuit Then, I = = Or Or Or E = IR + Ir E = V + Ir V = E - Ir Or r = Or r which is expression for internal resistance of a cell. Difference between emf and potential difference Emf Maximum value of potential difference between electrodes of a cell in open circuit is called emf It is independent of resistance of circuit. It is measured by potentiometer Potential Difference The potential difference between two electrodes of the cell in closed circuit i.e. when current is being drawn from it, is called terminal potential difference. It depends upone the resistance of the circuit. It is measured by voltmeter. GRUPING OF CELL 1. Cell in Series:
12 Let us consider n cells each of emf E and internal resistance r are connected in series, as shown in figure. Then, total emf of circit = E + E + E+ + E = ne And total internal resistance of circuit = r+ r + + r = nr Total resistance of circuit = nr + R I = = If R>>nr, then I = combination is useful. current in external circuit is n-times the current due to single cell. So, series If R<<nr, then I = combination is not useful in this case. current in external circuit is same as due to single cell. So, series 2. Cell in Parallel Let us consider m cells each of emf E and internal resistance r are connected in parallel, as shown in figure Then & E = total emf or circuit R = external resistance Then, total internal resistance is given by = = Or Total resistance of the circuit = R + = R + Or I = = If R<< r, then I = combination is useful. current in external circuit is n-times the current due to single cell. So, parallel If R>>r, then I = combination is not useful in this case. current in external circuit is same as due to single cell. So, parallel
13 Note in mixed grouping n-cells are connected in each series and m such combination are connected in parallel, as shown in figure Then, in the similar way, Total emf = ne Total resistance= R + Then, I = = Combination is useful when internal resistance is equal to external resistance. Note : Ohm s law is unable to give current in complicated circuits. Kirchoff s Law: Kirchoff s gave two laws which are very useful to give current in complicated circuits. Kirchoff s 1 st law or Junction Law: It states that, the algebraic sum of the currents at any unction in a circuit is zero. i.e. So, according to this law there is no accumulation of electric current at any junction (the points where wires meets in an electric circuit). By convention, the currents which flow towards a junction are taken as positive & the currents which flow away from the junction are taken as negative. From figure, we have- Kirchoff s 2 nd Law or Loop Law: it states that, in any closed circuit, the algebraic sum of the products of the current and resistance of each part of the circuit is equal to the total emf in the circuit. i.e.
14 By convention, the product of current and resistance is taken as positive when we move in the direction of flow of current. The emf is taken positive when we move from negative to positive terminal through the cell and vice-versa. Applying Kirchoff s to the circuit as shown in fig. For the loop ACDBA, For the loop EFDBA For the loop EFBAE ILLSTRATION: taking the current in the clockwise direction along ABCDA as positive 5I+ 10I + 8I + 10I + 5I = I = 40 I = A Wheatstone s Bride: It is based on Kirchoff s Law. It is consist of four resistances P,Q,R and S to form a closed path. A galvanometer G, a cell of emf E, a key K is connected as shown in figure. The current I is divided at various junctions according to Kirchoff s law as shown in figure. Now applying Kirchoff s Voltage law to the closed path ABDA.(1) Now applying Kirchoff s Voltage law to the closed path BCDB.(2) If bridge is balanced, then galvanometer will show no deflection i.e.
15 Therefore, equation(1) becomes.(3) And, equation (2) becomes.(4) Dividing (3) by (4) we have This is condition for bridge balance. If P,Q, and r are known the resistance S can be calculated. Slide Wire Bridge or Meter Bridge It is the practical application of wheatstone brdge. It is used to determine unknown resistance. Construction: it is consist of 100cm long wire stretched and clamped between two metallic strips bent at right angle as shown in figure.( between two points A and C ) The metallic strip has two gaps G 1 and G 2 where resistance P and Q are connected. A cell E is connected between A and C through a key K. here J is jockey connected to B through galvanometer G. This jockey can be move over wire. Working: A known resistance is taken out of the resistance box (Q) and jockey is moved till the deflection in the galvanometer is zero. When galvanometer shows no deflection (say at point J) the bridge is said to be balanced and therefore..(1) Here & Let, R = resistance of wire segment AJ S = resistance of wire segment JC r = resistance of the wire per unit length If AJ = l and JC = 100-l Then, R = r.l and S = r (100-l) Therefor, (2)
16 From (1) and (2) we have Or P = Hence by knowing Q and l we can calculate P Note: If P is Known and Q is unknown then Q = Potentiometer It is an instrument used to compare the emfs of two cells and to determine the internal resistance of a cell. Construction: It is consist of long uniform wire of constantan or manganin stretched in 4 to 10 segment, each of one metre length. Te segments are stretched parallel to each other on a wooden board. The ends of the wire are fixed to Cu strips. A metre scale is fixed parallel to the wire. To provide constant current through AB a source of emf E is connected across points A and B through an Ammeter a key K and a rheostat as shown in figure. Principle of Potentiometer An potential difference less than the total potential difference maintained across the potentiometer wire can be balanced against a convenient length of the potentiometer wire. The potential difference balanced is directly proportional to the length of the wire which balances the potential difference. Proof: Make the connections as shown in circuit diagram. Here E 1 > E. I is constnt current. G is galvanometer. If the potential difference between A and J is equal to the emf of the cell, no current flows through the galvanometer. It shows zero deflection. AJ = l is called balancing length. Then, potential difference across AJ = Irl Where, r= ressistance per unit length of potentiometer wire & I is current by E 1 So, E = Irl
17 Or E Irl, (As I and r are constant) Hence emf of the cell is directly proportional to its balancing length. i.e. The potential difference balanced is directly proportional to the length of the wire which balances the potential difference.which is principle of a potentiometer. To compare the emf s of two cells. If we want to compare the emf s of two cells, we make the connections as shown in the circuit diagram. Let r is the resistance per unit length of the potentiometer wire and I is the constant current flowing through it. Here is connected through a two way key. Now we insert key between gap PC will be out of circuit.) will be in circuit (and If balancing length in this case is l 1 then E 1 = l 1 ri (1) Now we insert key between gap QC will be in circuit (and will be out of circuit.) If balancing length in this case is l 2 then E 1 = l 2 ri (2) Dividing (1) by (2), we have- Or Hence by knowing we can compare emf s To determine the internal resistance of a cell: To determine the internal resistance of a cell E, we make the connections as shown in the circuit diagram. Here K is a key, if K is pressed E supply current which passes through resistance box RB.
18 Let r is the resistance per unit length of the potentiometer wire and I is the constant current flowing through it. Now let K is not pressed, cell E will not supply any current. If balancing length in this case is l 1 then, emf of the cell will be- E = l 1 ri (1) Now let K is pressed, cell E will not supply any current. If balancing length in this case is l 2 then, terminal potential difference of the cell will be Dividing (1) by (2), we have- V = l 2 ri (2) Or As the internal resistance r 1 of the cell is given by r 1 = Therefore, r 1 = Hence knowing we can calculate the internal resistance of a cell. NOTE : FOR MORE DETAIL CONSIDER NCERT BOOKS. DOWNLOAD NCERT BOOKS FORM
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