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![G K GEORGE Page 11
When K is closed, 𝐼 =
𝐸
𝑅+𝑟
𝐸 = 𝐼𝑅 + 𝐼𝑟
Since R is parallel to electrodes of the cell, so the terminal potential difference
of the cell is equal to the potential difference across R
i.e. V = IR
Therefore, 𝐸 = 𝑉 + 𝐼𝑟
i.e Terminal potential difference is less than EMF ( TPD will be more than EMF
when the battery is charging)
Internal Resistance: -
𝐼 =
𝐸
𝑅 + 𝑟
𝑉 =
𝐸𝑅
𝑅+𝑟
, 𝑅 + 𝑟 =
𝐸𝑅
𝑉
Or 𝑟 = [
𝐸
𝑉
− 1] 𝑅
Grouping of cells:-
1) Series grouping:
Consider n identical cells, each of emf E and internal resistance r connected in
series to an external resistance R](https://image.slidesharecdn.com/c-231011153348-d4430cbf/75/C-12-PHYSICS-CURRENT-ELECTRICITY-notes-pdf-11-2048.jpg)
![G K GEORGE Page 17
Principle:
When a constant current is passed through a wire of uniform area of cross
section; the potential drop across any portion of the wire is directly
proportional to the length of that portion.
It consists of a uniform resistance wire
There is a jokey which can be moved along the wire to make any contact
with it at any point
A cell of emf E connects in series with potentiometer wire forms the
primary circuit
Let V be the potential difference across any portion of length 𝒍,
resistance R and uniform area of cross section A. if 𝑰 be the current flowing
through the wire.
V = IR =
𝑰𝝆𝒍
𝑨
= [
𝑰𝝆
𝑨
] 𝒍 = 𝒌𝒍
i.e. V α𝒍, provided I,ρ and A are constants.
Comparison of emf’s of 2 cells: -
The positive terminal of the cell is connected to A and the negative
terminal is connected to B through rheostat and key. The cells whose emfs to
be compared are connected by using keys a and b. The key a is adjusted in such
a way that E1 is included in the circuit. Balancing length is measured as 𝒍𝟏 by
null deflection method. Then key b is adjusted for E2. So balancing length is 𝒍𝟐.](https://image.slidesharecdn.com/c-231011153348-d4430cbf/75/C-12-PHYSICS-CURRENT-ELECTRICITY-notes-pdf-17-2048.jpg)
![G K GEORGE Page 18
𝑬𝟏 ∝ 𝒍𝟏 ; 𝑬𝟐 ∝ 𝒍𝟐
𝑬𝟏 = 𝒌𝒍𝟏; 𝑬𝟐 = 𝒌𝒍𝟐
𝑬𝟏
𝑬𝟐
=
𝒍𝟏
𝒍𝟐
Determination of internal resistance of a cell by using Potentiometer: -
Step-1
K1 is closed and K2 is open. At the point S, jokey give null deflection.
𝑬 ∝ 𝒍𝟏; 𝑬 = 𝒌𝒍𝟏
Step-2
K2 is closed so that the known resistance is connected across the cell.
Find the null deflection. The terminal potential difference 𝑽 ∝ 𝒍𝟐
𝑽 = 𝒌𝒍𝟐
𝑬
𝑽
=
𝒍𝟏
𝒍𝟐
Internal resistance, 𝑟 = [
𝐸
𝑉
− 1] 𝑅
𝑟 = [
𝑙1
𝑙2
− 1] 𝑅
*notes printed in red colour excluded from the syllabus for the academic
session 2023-24](https://image.slidesharecdn.com/c-231011153348-d4430cbf/75/C-12-PHYSICS-CURRENT-ELECTRICITY-notes-pdf-18-2048.jpg)
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C.12 PHYSICS CURRENT ELECTRICITY notes.pdf
- 1. G K GEORGEPage 1 CURRENT ELECTRICITY Electric current: - It implies some sort of motion It denotes the motion of electric charge Electric current produced by motion of charge carriers in medium like electrons in metal, ions in liquids and gases It is either due to flow of positive or negative charge Direction of current is taken as the direction of positive charge or opposite to negative charge This will continue until the potential become equal Electric current can be defined as amount of charge flowing through any cross section of a substance in a unit time Or It is the rate of flow of charge through any cross section of a substance. 𝑰 = 𝑸 𝒕 If n carriers of electricity each having charge e, then 𝑰 = 𝒏𝒆 𝒕 Unit:- Ampere ( A ) The current through a wire is called one ampere, if one coulomb of charge flows through the wire in one second. Types of current: - Steady direct current An electric current is said to be steady direct current if its magnitude and direction do not change with time.
- 2. G K GEORGEPage 2 Varying or variable direct current An electric current is said to be varying direct current if its magnitude change with time and polarity remains same Alternating current An electric current is said to be alternating current if its magnitude changes with time and polarity reverses periodically. Current density: - Current density of a conductor is defined as the amount of current flowing per unit area of the conductor held perpendicular to the flow of current. Current density= J = current / area, unit is A /m2 It is a vector quantity. 𝑑𝐼 = 𝐽 ⃗ ∙ 𝑑𝑠 ⃗⃗⃗⃗⃗ 𝐼 = ∫ 𝐽 ⃗∙ 𝑑𝑠 ⃗⃗⃗⃗⃗ 𝑠 Current is a scalar quantity
- 3. G K GEORGEPage 3 Current in a metallic conductor: - Free electrons are continuously moving about within the metal, with different velocities in different directions During their travel, they collide with atoms (ions) gaining and losing kinetic energy. Since motion is random, the average thermal velocity of electrons is zero. As a result, there is no net flow of charge in any direction. When potential difference applied across the conductor, an electric field (𝑬 ⃗⃗⃗) established in the conductor. Due to this electric field, force experience on electron opposite to the electric field (because electric field from positive to negative). Due to this force, the electron acquired an acceleration 𝒆𝑬 𝒎 The acceleration will last only for short time, since electrons are deflected by positive ions. Between successive collisions an electron acquired a velocity component opposite to electric field. If 𝝉 represents average time between successive collisions, the drift velocity𝒗 = 𝒆𝑬 𝒎 𝝉 Or 𝒗 ⃗ ⃗⃗ = −𝒆𝑬 ⃗⃗⃗ 𝒎 𝝉 The average additional velocity of free electrons in a conductor subjected to an electric field is called Drift velocity.
- 4. G K GEORGEPage 4 RESISTANCE AND RESISTIVITY: Ohm’s law: The electric current I flowing through the substance is proportional to the voltage across their ends. V α I V = RI “When temperature is constant, the current flowing through a conductor is directly proportional to the potential difference between the ends of the conductor “ Resistance is the ratio of the potential difference applied across the conductor to the current flowing through it. R = V/I Unit- ohm(Ω) Dimension- ML2 T-3 A-2 Resistance of a conductor is said to be one ohm, if one ampere of current flows through it, when a potential difference of one volt is applied across it. Resistivity: - At constant temperature, the resistance of the conductor is directly proportional to the length and inversely proportional to the area of cross section.
- 5. G K GEORGEPage 5 𝑅 α 𝑙 𝐴 , i.e. 𝑅 = 𝜌𝑙 𝐴 𝜌 = 𝑅𝐴 𝑙 If A= 1m2 and l=1m, then ρ = R Resistivity of the material of a conductor is defined as the resistance of the conductor of unit length and unit area of cross section. Unit- ohm-metre ( Ω ) Dimension- ML3T-3A-2 The reciprocal of resistivity is called Conductivity 𝜎 = 1 𝜌 = 𝑙 𝑅𝐴 Color code: - First two ------ significant figures Third ----------- multiplayer Forth ---------- tolerance pq X 10r ±𝒔% B B ROY Great Britain Very Good Wife
- 6. G K GEORGEPage 6 Origin of resistivity: - Consider a conductor of length ’l’ and area of cross section A. Let n be the electron density. When a voltage V is applied across its ends, a uniform electric field E is established inside. The magnitude of electric field 𝐸 = 𝑉 𝑙 If v be the drift velocity of electron, then the number of electrons passing through the conductor per second is nAv Therefore, the charge flowing through the conductor per second is nAve I = nAve,V = IR 𝑬𝒍 = 𝒏𝑨𝒆𝒗𝑹 𝑬 = 𝒏𝒆𝒗𝑨𝑹 𝒍 = nev𝝆 But v = eEƮ / m
- 7. G K GEORGEPage 7 𝒗𝒎 𝝉𝒆 = 𝒏𝒆𝒗𝝆 or𝝆 = 𝒎 𝒏𝝉𝒆𝟐 ρ is related to n and collision time. The collision time depends on vibration of atom of the metal. As temperature increases, the amplitude of vibration of the atom increases. This lead to more collision of electrons with atom; thus reducing the value of Ʈ, so the resistance of the conductor increases with temperature. Mobility: - It is defined as the magnitude of the drift velocity per unit electric field. 𝜇 = |𝑣𝑑| 𝐸 𝒗𝒅 = 𝒒𝑬 𝒎 𝝉 , 𝜇 = 𝑞𝝉 𝑚 Temperature dependence on resistivity: - Resistivity increases with temperature since resistance is directly proportional to resistivity. Let R1 be the resistance at t1 0C and R2 at t2 0C (t2> t1) The increase in resistance = R2 – R1 R2 – R1αR1 α(t2- t1) i.e. R2 – R1αR1(t2- t1) i.e. R2 – R1= 𝛼R1(t2- t1) or α = R2 – R1 R1(t2 − t1) If t1 = 0 0 C and R1 =R0 t2 = t 0 C and R2 =R1
- 8. G K GEORGEPage 8 then, α = 𝑅1−𝑅0 𝑅0𝑡 α = 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒 𝑖𝑛 𝑟𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑟𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑎𝑡 00𝐶 𝑋 𝑟𝑖𝑠𝑒 𝑖𝑛 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒 The temperature coefficient of resistance is defined as the change in resistance per unit original resistance per degree rise in temperature Or The ratio of increase in resistance for unit degree rise of temperature to its resistance at 0 0C Resistance of the material generally increases with temperature. i.e. they have positive temperature coefficient. Eg:-Aluminium, copper, brass etc. Resistance of certain material does not vary with temperature. i.e. they have zero temperature coefficient. Eg:- Manganin Resistance of certain material decreases with temperature. i.e. they have negative temperature coefficient. Eg:- carbon, germanium and silicon. Limitations of Ohm’s law: - V depends on I non-linearly This happens when the current through the conductor is very large. The large current heats the conductor H = I2 Rt, thereby increases its resistance and thereby decreases the current. i.e. voltage – current relation is linear only for small currents The relation between V and I depend on the sign of V in addition to its magnitude. When voltage V is applied across p-n junction, current obtained only when the p-n junction is forward biased
- 9. G K GEORGEPage 9 The relation between V and I is non-unique. This happens at low temperature. It is found that at low temperature ( less than 4K), the resistance of wire of length 2𝑙 𝑖𝑠 𝑔𝑟𝑒𝑎𝑡𝑒𝑟 𝑡ℎ𝑎𝑛 𝑡𝑤𝑖𝑐𝑒 𝑡ℎ𝑒 𝑟𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒 of the same wire of length 𝑙 COMBINATION OF RESISTORS: 1) SERIES COMBINATION In this case the current remains same and voltage different across the resistors V1 = IR1; V2 = IR2; V3 = IR3 IR = IR1 + IR2 + IR3 R = R1 + R2 + R3 2) PARALLEL COMBINATION
- 10. G K GEORGEPage 10 In this case the current different and voltage remains same across the resistors I = I1 + I2 + I3 𝑽 𝑹 = 𝑽 𝑹𝟏 + 𝑽 𝑹𝟐 + 𝑽 𝑹𝟑 𝟏 𝑹 = 𝟏 𝑹𝟏 + 𝟏 𝑹𝟐 + 𝟏 𝑹𝟑 EMF:- It is defined as the potential difference between the terminals of a cell in an open circuit. In a conductor, the positive charge will flow in the direction of electric field. To maintain a steady current, the conductor cannot be isolated, it must be part of a closed circuit, that include an external device. This device is required to transport positive charge from low potential to high potential, thus maintain a potential difference between the two points. The external device will need to do work for it. Such a device is the source of EMF. The work done by the cell in forcing unit positive charge to flow in the whole circuit (includingcell) is called the emf of the cell. Terminal potential difference: - Terminal potential difference of a cell is defined as the potential difference between its terminals in a closed circuit. Internal resistance: - It is defined as the opposition offered by a cell to the flow of current through it. It is denoted by r. It depends on nature of the electrolyte and electrodes. Relation between EMF and Terminal potential Difference: -
- 11. G K GEORGEPage 11 When K is closed, 𝐼 = 𝐸 𝑅+𝑟 𝐸 = 𝐼𝑅 + 𝐼𝑟 Since R is parallel to electrodes of the cell, so the terminal potential difference of the cell is equal to the potential difference across R i.e. V = IR Therefore, 𝐸 = 𝑉 + 𝐼𝑟 i.e Terminal potential difference is less than EMF ( TPD will be more than EMF when the battery is charging) Internal Resistance: - 𝐼 = 𝐸 𝑅 + 𝑟 𝑉 = 𝐸𝑅 𝑅+𝑟 , 𝑅 + 𝑟 = 𝐸𝑅 𝑉 Or 𝑟 = [ 𝐸 𝑉 − 1] 𝑅 Grouping of cells:- 1) Series grouping: Consider n identical cells, each of emf E and internal resistance r connected in series to an external resistance R
- 12. G K GEORGEPage 12 Eeff = E + E + E + E + ……….. = nE reff = r + r + r + …………….. = nr Total resistance = R + nr Current I = effective emf / equivalent resistance I = nE / ( R + nr ) Special case:- i) R >> nr I = n ( E/R) In order to get a large amount of current from the cells connected in series; the external resistance will be very large as compared to effective internal resistance. ii) R<<nr; I= E/r is the current due to single cell. 2) Parallel grouping: - Consider m identical cells each of emf E and internal resistance r.
- 13. G K GEORGEPage 13 Total emf = E 1/reff = 1/r + 1/r +………………..= m/r Therefore I = E / (R + r/m) = mE/ (mR + r) Special case:- i) If R>>r; R+r/m =R I = E/R, current due to single cell. ii) If R<<r; R+r/m =r/m, then I = mE/r. In order to get large current, the cells may be connected in parallel to a small external resistance. Mixed grouping of cells: Eeff = nE 1/reff = 1/nr + 1/nr + ………………=m/nr 𝑰 = 𝒏𝑬 𝑹 + 𝒏𝒓 𝒎 = 𝒏𝒎𝑬 𝒎𝑹 + 𝒏𝒓 The current in the circuit will be maximum if mR + nr = minimum 𝒎𝑹 + 𝒏𝒓 = {√𝒏𝒓 − √𝒎𝑹} 𝟐 + 𝟐√𝒎𝒏𝒓𝑹 mR + nr = minimum means the quantity √nr − √mR is minimum. It is a square term, so never be negative. So the least possible value is zero. √𝒏𝒓 = √𝒎𝑹 i.e. nr = mR or R = nr/m
- 14. G K GEORGEPage 14 Thus, in order to get the maximum current in the circuit, the mixed grouping of cells must be done in such a way that the external resistance is equal to the effective internal resistance of the cells. KIRCHHOFF’S LAWS: - 1) Current law or junction law It states that the sum of all the currents entering in any point (junction) must be equal to the sum of all the currents leaving that point (junction). Or The algebraic sum of all the currents meeting at a point (junction) in a closed electrical circuit is zero. I1 + I2- I3- I4- I5=0 The current entering to the point is taken as positive and while current leaving the point is taken as negative. 2) Voltage law or Loop law: - It states that the algebraic sum of the products of the currents and resistances of any closed loop is equal to the total emf in that loop. ∑ 𝑰𝑹 = ∑ 𝑬
- 15. G K GEORGEPage 15 In loop ABCFA, I1R1-I2R2 = E1-E2 In loop CDEFC, I2R2 +( I1+I2)R3 = E2 In loop ABCDEFA, I1R1+(I1+I2)R3 = E1 WHEATSTONE BRIDGE:- It is an arrangement of 4 resistances used for measuring one unknown resistance in terms of the other 3 known resistances Principle:- When key KG is closed, galvanometer G shows deflection, i.e. the presence of Ig. The value of R is adjusted in such a way that galvanometer shows no deflection. At this stage the potential at points B and D are equal. The wheatstone bridge is balanced at this stage.Then, 𝑃 𝑄 = 𝑅 𝑆 OR 𝑆 = ( 𝑄 𝑃 ) 𝑅 Proof:- At the junction A, the current I divide into I1 through P and ( I-I1) through R. Applying Kirchhoff’s law to the loop ABDA I1P + IgRg - ( I - I1 )R = 0 I1P + IgRg = ( I - I1 )R------------------------(1) For the loop BCDB
- 16. G K GEORGEPage 16 ( I1-Ig)Q - ( I - I1 + Ig)S -IgRg= 0 ( I1-Ig)Q = ( I - I1 + Ig)S+IgRg------------------------(2) When bridge is balanced, Ig = 0. So (1) and (2) becomes I1P = ( I - I1 )R------------------------(3) I1Q = ( I - I1 )S------------------------(4) (3)/(4), 𝑃 𝑄 = 𝑅 𝑆 Metre bridge:- It consists of one metre resistance wire. The ends of the wire connected to two thick copper strips. A third copper strip is placed between the above, providing two gaps G1 and G2. The resistance box is connected at G1 and the resistance to be measured is connected to G2. Close the key and adjust the known value of resistance R from the resistance box. Now move jokey(J) over the wire AC so that the galvanometer shows no deflection. Now the bridge is said to be balanced. Let the resistance between A and B = P the resistance between B and C = Q If r be the resistance per unit length, then 𝑃 = 𝑟𝑙, 𝑎𝑛𝑑 𝑄 = (100 − 𝑙)𝑟 𝑺 = ( 𝑸 𝑷 ) 𝑹 = (𝟏𝟎𝟎 − 𝒍)𝒓𝑹 𝒍𝒓 = (𝟏𝟎𝟎 − 𝒍)𝑹 𝒍 Potentiometer: -
- 17. G K GEORGEPage 17 Principle: When a constant current is passed through a wire of uniform area of cross section; the potential drop across any portion of the wire is directly proportional to the length of that portion. It consists of a uniform resistance wire There is a jokey which can be moved along the wire to make any contact with it at any point A cell of emf E connects in series with potentiometer wire forms the primary circuit Let V be the potential difference across any portion of length 𝒍, resistance R and uniform area of cross section A. if 𝑰 be the current flowing through the wire. V = IR = 𝑰𝝆𝒍 𝑨 = [ 𝑰𝝆 𝑨 ] 𝒍 = 𝒌𝒍 i.e. V α𝒍, provided I,ρ and A are constants. Comparison of emf’s of 2 cells: - The positive terminal of the cell is connected to A and the negative terminal is connected to B through rheostat and key. The cells whose emfs to be compared are connected by using keys a and b. The key a is adjusted in such a way that E1 is included in the circuit. Balancing length is measured as 𝒍𝟏 by null deflection method. Then key b is adjusted for E2. So balancing length is 𝒍𝟐.
- 18. G K GEORGEPage 18 𝑬𝟏 ∝ 𝒍𝟏 ; 𝑬𝟐 ∝ 𝒍𝟐 𝑬𝟏 = 𝒌𝒍𝟏; 𝑬𝟐 = 𝒌𝒍𝟐 𝑬𝟏 𝑬𝟐 = 𝒍𝟏 𝒍𝟐 Determination of internal resistance of a cell by using Potentiometer: - Step-1 K1 is closed and K2 is open. At the point S, jokey give null deflection. 𝑬 ∝ 𝒍𝟏; 𝑬 = 𝒌𝒍𝟏 Step-2 K2 is closed so that the known resistance is connected across the cell. Find the null deflection. The terminal potential difference 𝑽 ∝ 𝒍𝟐 𝑽 = 𝒌𝒍𝟐 𝑬 𝑽 = 𝒍𝟏 𝒍𝟐 Internal resistance, 𝑟 = [ 𝐸 𝑉 − 1] 𝑅 𝑟 = [ 𝑙1 𝑙2 − 1] 𝑅 *notes printed in red colour excluded from the syllabus for the academic session 2023-24