### Heating Effect on Electric Currents

The heat in electric cells is caused because of inelastic collisions of electrons with the surrounding atoms, known as lattice atoms, which receive the kinetic energy and vibrate, thus increasing temperature.

**Circuit diagram**

**Electrical Resistance (R)**

- Definition: “The potential difference V across a component divided by the current passing through it.”. R = V/I.
- Units: Ohms (Ω). 1 Ω = 1 V A^-1
- Ohm’s law: “At constant temperature, the voltage across a component is proportional to the current passing through it”.
- Graph: Resistance can be determined as V/I in any point of the graph’s line.

Non-ohmic components: Components that do not obey Ohm’s law.

- Lamp bulb/filament lamp: ohmic at low currents, because high currents cause great temperature increase, causing resistance to rise, and thus, current to decrease.

- Semiconducting diodes: only allow flow in one direction.

- Thermistor: as the temperature increases, its resistance falls, as the lattice ions vibrate more and impede charge-carriers movement.
- Resistivity (ρ) = RA/l, where R is the resistance, A is the area and l is the length of the wire.
- Combining resistors:

Electrical Power (P)

Power = Work done/Time taken = qV/t = IV = RI^2 = V^2/R

Potential divider

- Definition: circuit component that changes the voltage according to each specific situations.
- Using sensors: one fixed-value resistor and one sensitive resistor (to external conditions).
- Thermistor: negative temperature coefficient: resistance proportional to 1/temperature.
- Light-dependent resistor (LDR): resistance proportional to 1/light intensity.

- Potentiometer (rheostat): allows a wide range of potential differences, depending on the connection point of the slider, giving a max-value emf, an advantage of series of resistors.

### Kirchhoff’s circuit laws

- First law: “The sum of the currents/total charge flowing into a junction equals the sum of the currents/total charge flowing away from a junction.” ∑I = 0.
- Second law: “In a complete circuit loop, the sum of the voltages equals zero.” ∑∆V = ∑IR = 0.

**Measuring devices**

- Ammeter: Measures the current of a circuit. In series with the circuit, with ideal zero resistance. Non-ideal ammeters have low constant resistance.
- Voltmeter: Measures the potential difference across a device. In parallel with the circuit, with ideal infinite resistance. Non-ideal voltmeters have high constant resistance.

### 5.3 Electric Cells

Electromotive force (emf/ε): “Total potential difference across a cell’s terminals when no current is flowing (I = 0).” ε = W/q = P/I = ∑potential differences in the circuit.

Internal resistance (r): “Resistance of the components/chemicals within the cell itself, leading to energy/power loss in the cell.” It is possessed by a real battery. ε = I(R+r).

Terminal potential difference: pd across a real battery’s terminals, V = ε – IR.

**V-I graph for a real cell:**

Cells

- Battery: chemical energy transformed into thermal, mechanical,…energy.
- Primary cell (Non-rechargeable):”cells used until they are exhausted and then discarded”.
- Secondary cell (rechargeable):”possibility of the reversion of chemical reactions into original form”. e.g. lead-acid cell.
- Recharging process: Return the energy in the reverse current direction at dp above nominal.

- Discharging a cell: “The terminal potential difference of a typical practical electric cell loses its initial value quickly, has a stable and constant value for most of its lifetime, followed by a rapid decrease to zero as the cell discharges completely” (IB Physics Guide, 2014).

### About this unit

Electric current, flow of electric charges in a metallic conductor, drift velocity and mobility, and their relation with electric current; Ohm’s law, electrical resistance, V-I characteristics (liner and non-linear), electrical energy and power, electrical resistivity and conductivity. Carbon resistors, colour code for carbon resistors; series and parallel combinations of resistors; temperature dependence of resistance. Internal resistance of a cell, potential difference and emf of a cell, combination of cells in series and in parallel. Kirchhoff’s laws and simple applications. Wheatstone bridge, metre bridge.Potentiometer-principle and applications to measure potential difference, and for comparing emf of two cells; measurement of internal resistance of a cell

### CURRENT ELECTRICITY

### ELECTRIC CURRENT

- Current carriers in conductor are electrons, (valence e–s) ions in electrolytes, electrons & holes in semiconductor and positive ions /electrons in gases.
- Charge of electron = 1.6 × 10–19c
- 1 ampere = 6.25 × 1018 electrons/sec
- Though direction is associated with current (opposite to the motion of electrons), but it is not a vector quantity as it does not follow rules of vector addition.
- For a current to flow through a cross-section, there must be a net flow of charge through that cross-section.

In a metal like copper there are around 1028 free electrons per m3 moving randomly in all direction with speeds of the order of 106 m/s even in the absence of electric field. But since the number of electrons passing through a cross-section from left to right is equal to the number of electrons passing from right to left in a given time, therefore the net charge flow is zero and hence the electric current is zero. - A conductor remains uncharged when current flows in it. i.e. Net charge in a current carrying conductor is zero.

### CURRENT DENSITY

- Electric current is a macroscopic physical quantity where as current density is a microscopic physical quantity.
- For a given conductor current does not change with change in cross-sectional area.

### DRIFT VELOCITY _{}

A = area of cross-section, vd = drift velocity

- Drift velocity is very small, it is of the order of 10–4m/s which is negligible as compared to thermal speed of e–s at room temperature (105 m/s)
- The drift velocity is given by

- where, J = current density
- e = electronic charge = 1.6 × 10–19 C
- n = the number of free electrons per unit volume

- The number of free electrons per unit volume (n) can be determined by the following relation :

- For steady current :

- Variation of drift velocity :

### OHM’S LAW AND ELECTRICAL RESISTANCE

#### ELECTRICAL RESISTANCE

#### RESISTIVITY

#### CONDUCTIVITY

- The value of ρ is very low for conductor, very high for insulators & alloys, and in between those of conductors & insulators for semiconductors.
- Resistance is the property of object while resistivity is the property of material.

#### EFFECT OF TEMPERATURE ON RESISTANCE AND RESISTIVITY

- in making very strong electromagnets
- to produce very high speed computers
- in transmission of electric power
- in the study of high energy particle physics and material science

### SERIES AND PARALLEL COMBINATION OF RESISTORS

#### RESISTANCES IN SERIES

#### RESISTANCES IN PARALLEL

### HOW TO FIND EQUIVALENT RESISTANCE?

#### SUCCESSIVE REDUCTION

#### AXIS SYMMETRY

#### SHIFTED SYMMETRY

#### PATH SYMMETRY

#### STAR-DELTA CONNECTION

- Resistors are not just in series or in parallel if they look so geometrically, e.g. the resistors in the diagram are not in parallel but in series.

- This is a common thinking that current which comes out from the positive terminal of a battery is used up till it reaches the negative terminal. But infact the current remains the same in a branch. In fact a potential drop takes place across a resistor.

- 🗴 Incorrect : If two resistances are not in series then it is in parallel and vice-versa.

### COLOUR CODING FOR CARBON RESISTOR AND THEIR STANDARD VALUES

- The first and second colour bands, represent the first and second significant digits respectively, of the resistance value.
- The third colour band is for the number of zeros that follow the second digit.
- In case the third band is gold or silver, it represents a multiplying factor of 0.1 or 0.01.
- The fourth band represents the manufacturer’s tolerance. It is a measure of the precision with which the resistor was made.
- If the fourth band is not present, the tolerance is assumed to be ± 20%.

- Wire round resistor made by winding of wires of an alloy manganins, constantan and nichrome.
- Carbon resistors have low cost and are compact.

### THERMISTOR

### JOULE’S LAW OF HEATING

- square of the current flowing through the conductor,

(q, T – constt) i.e. H ∝ i2 - resistance of the conductor (i, T – constt.)

- time for which the current is passed (i, R, – constt)

### ELECTRIC POWER

### ELECTROMOTIVE FORCE AND INTERNAL RESISTANCE OF A CELL

- Electromotive force is not a force but a potential difference.
- E.m.f. can be defined as the work done in moving a charge once around a closed circuit.

#### INTERNAL RESISTANCE (r)

- For a cell

- Emf is the property of a cell but terminal potential difference depends on the current drawn from the cell.

#### SHORT CIRCUITING

### COMBINATION OF CELLS

#### SERIES COMBINATION OF CELLS

EAB = E1 + E2 + … + En

RAB = r1 + r2 + ……. + rn

#### PARALLEL COMBINATION OF CELLS

#### MIXED GROUPING OF CELLS

- The condition for maximum current through external resistance R

- Maximum power dissipation for the circuit shown in fig.

- If identical cells are connected in a loop in order, then emf between any two points in the loop is zero.
- If n identical cells are connected in series and m are wrongly connected then

Enet = nE – 2mE

### FARADAY’S LAW OF ELECTROLYSIS

- 1st law : The mass of the substance liberated or deposited at an electrode during electrolysis is directly proportional to the quantity of charge passed through the electrolyte.

- 2nd law : When the same amount of charge is passed through different electrolytes, the masses of the substance liberated or deposited at the various electrodes are proportional to their chemical equivalents

#### FARADAY’S CONSTANT

- If ρ is the density of the material deposited and A is the area of deposition, then the thickness (d) of the layer deposited in electroplating process is .
- The back e.m.f. for water voltameter is 1.67 V and it is 1.34 V for CuCl2 electrolytes voltameter with platinum electrodes.
- 96500 C are required to liberate 1.008 g of hydrogen.
- 2.016 g of hydrogen occupies 22.4 litres at N.T.P.
- E.C.E. of a substance = E.C.E. of hydrogen × chemical equivalent of the substance.

### SEEBECK/THERMOELECTRIC EFFECT

### PELTIER EFFECT

### THOMSON EFFECT

- The actual emf developed in a thermocouple loop is the algebraic sum of the net Peltier emf and the net Thomson emf developed in the loop.

- If S, π and σ are the Seebeck coefficient, Peltier coefficient, and Thomson coefficient respectively then it is found that
- For Peltier effect or Thomson effect, the heat evolved or absorbed is directly proportional to current. But for Joule’s law of heating, the heat produced is directly proportional to the square of the current flowing through it.
- Thermo-emf set up in a thermocouple when its junctions are maintained at temperature T1 and T3 (i.e. ) is equal to the sum of the emfs set up in a thermocouple when its junctions are maintained first at temperature T1 and T2 (i.e. ) and then at T2 & T3 (i.e. ) i.e.

### KIRCHOFF’S LAWS AND ELECTRICAL CIRCUIT

#### KIRCHOFF’S JUNCTION LAW

#### KIRCHOFF’S LOOP LAW

If we move a loop element (resistor, emf device, capacitor, inductor etc.) in the direction of increasing potential, we take the potential difference positive and vice-versa.

#### PROBLEM SOLVING TACTIC FOR USING KIRCHOFF’S LAW

- Draw a circuit diagram large enough to show all resistors, emf device, capacitors, currents clearly.
- Take into account the resistance of voltmeter/ammeter/internal resistance of a cell (if given).
- Assume the direction of current in all branches. It may be noted here that one branch has only one direction of current. It is best to use junction law simultaneously while drawing currents. This helps to reduce the number of unknown quantities.

- In a branch containing a capacitor, the current is zero when d.c is applied and steady state conditions are achieved.
- Now we need as many independent equations as there are conditions unknowns. If we have to find a particular unknown, we should ensure that, the unknown appears in one of the equations made by us.
- For making equations choose the loop and travel the loop completely. We may travel the loop in clockwise or anti-clockwise direction. While using second law use sign conventions properly.
- Solve the equations formed to find the unknown quantities. If any value of current comes out to be negative then that particular current is in the opposite direction to that assumed.

#### APPLICATIONS

#### NODE METHOD TO APPLY KIRCHOFF’S LAW (OPEN LOOP METHOD)

### WHEATSTONE BRIDGE

### METER BRIDGE OR SLIDE WIRE BRIDGE

### POTENTIOMETER

- IB > IE. This happens when VPC > E. One side deflection in galvanometer
- IB = IE. This happens when VPC = E, Zero deflection in galvanometer
- IB < IE . This happens when VPA < E. Other side deflection in galvanometer

- At null point since no current flows through E therefore it is said to be in the condition of open circuit.
- More is the length of potentiometer, higher is the sensitivity of potentiometer and smaller is the potential gradient.
- Potentiometer will work only when B > E. Also the positive terminal of the batteries is connected at P. If any of the above conditions is not followed, we do not get a null point.

- Comparison of emfs of cells

- To find internal resistance of a cell
- Emf can be measured by potentiometer and not voltmeter.

### MEASURING INSTRUMENTS

#### GALVANOMETER

#### AMMETER

#### VOLTMETER