Electric Resistance

 Ohm's Law:

Ohm’s law formula is written as;

V  I

Therefore, V = RI, where R is a constant called resistance. R depends on the dimensions of the conductor and also on the material of the conductor. Its SI unit is Ohm (Ω)

The dimensional formula of resistance is given by,

M¹ L² T⁻³ I⁻²

Ohm’s law states that

" The current through a conductor between two points is directly proportional to the voltage across the two points"

 

Vector Form of Ohm’s Law:

The vector form is given as,

 Where,

J= Current Density

E=Electrical Conductivity

Example 1: If the resistance of an electric iron is 50 Ω and a current of 3.2 A flows through the resistance. Find the voltage between two points.

Solution:

 We use the following formula to calculate the value of V:

V = I × R

Substituting the values in the equation, we get

V = 3.2 A × 50 Ω = 160 V

Limitations of Ohm’s Law

« Ohm’s law is not applicable for unilateral electrical elements like diodes and transistors as they allow the current to flow through in one direction only.

« For non-linear electrical elements with parameters like capacitance, resistance etc the ratio ofvoltage and current won’t be constant with respect to time making it difficult to use Ohm’s law.

Ohm’s Law Applications

The main applications of Ohm’s law are:

To determine the voltage, resistance or current of an electric circuit.

Ohm’s law maintains the desired voltage drop across the electronic components.

Ohm’s law is also used in DC ammeter and other DC shunts to divert the current.

Electrical Conductivity:

Electrical Conductivity is an intrinsic property of a material which is defined as the measure of the amount of electrical current a material can carry.

Electrical conductivity is also known as specific conductance, and

 The SI unit is Siemens per meter (S/m).

It is also defined as the ratio of the current density to the electric field strength.  It is represented by the Greek letter σ.

Electrical Resistivity:

Electrical resistivity is the reciprocal of electrical conductivity. It is the measure of the ability of a material to oppose the flow of current.

Resistivity Unit:

 Resistivity Formula:

 Conductors with a uniform cross-section and uniform flow of electric current will have the following resistivity formula:

Where,

ρ is the resistivity of the material in Ω.m

R is the electrical resistance of uniform cross-sectional material in Ω

l is the length of a piece of material in m

A is the cross-sectional area of the material in m2

Factors Determining Resistivity of Materials:

Following are the factors that determine the resistivity of materials:

·        Nature of material

·        Temperature

The Resistivity of Different Materials:

·        Metals are good conductors of electricity. Hence, they have low resistivity.

·        The insulators like rubber, glass, graphite, plastics, etc. have very high resistivity when compared to the metallic conductors.

               The third type is a semiconductor, which lies between  conductors and insulators. Resistance decreases with increasing temperature and is also affected by the presence of impurities in them.

·        The table below lists the electrical resistivity of several conductors, semiconductors, and insulators.

 

Power Dissipated across Resistor:

Power dissipation is the process by which power is lost as heat as a result of  primary activity. This is a natural process. All  resistors that are part of a circuit and have a voltage drop across them  dissipate power.

Power Dissipated in a Resistor

The equation of power is P = v x i

P = i2R [since v = iR]

Where

 v is the voltage.

i is the current.

R is the resistance.

1) What is the power dissipated in a 10kΩ resistor when a current of 5mA passes through the resistor?

Answer:

                Power, P = i2R

                 (5 x 10-3)2 x 10 x 103

                 = 250 mW

The power dissipated in the resistor is 250 mW.

Temperature Dependence on Electrical Resistance :

The  resistance of a conductor is affected by temperature, so the resistance value also affects the rise in temperature. Temperature and conductor resistance are inversely proportional. The  resistance of the conductor increases with increasing temperature. Changes in the resistance of a conductor due to temperature increase the  resistance. As the  physical dimensions of a conductor change, so does its resistance. The resistance of conductors correlates inversely  with  area and inversely  with length. Thus, a certain wire length and diameter corresponds to a certain resistance value. Therefore, conductive materials tend to become more resistant as the temperature increases. In contrast, the resistance of the insulating material decreases as the temperature increases.

 Resistor Colour Code:

The colour bands are printed on them to represent the electrical resistance. These colour bands are known as resistor colour codes. The resistor colour code was invented in the 1920s by the Radio Manufacturers Association (RMA)

Several bands  specify the resistance value, the tolerance rate and sometimes the reliability or failure rates. The number of bands present in a resistor varies from three to six. The first two bands indicate the resistance value and the third band serves as a multiplier. In this piece of article.

Variation of Temperature with Resistivity:

Based on the conductivity of the materials, they are classified into three – conductors, semiconductors, and insulators. Conductors have low resistivities ranging from 10-8 Ω m to 10-6 Ω m while insulators have high resistivities which can be 1018 times greater than metals. Resistivity is indirectly proportional to the temperature. If we increase the temperature of materials, their resistivities will decrease. But this is not true for every material i.e., all materials do not have the same dependence on temperature.

The resistivities of metallic conductors within a limited range of temperatures are given by the following equation:

ρT= ρ0 [1 + a(T–T0)]

Here, ρT = resistivity at a temperature T

ρ0 = resistivity at a reference temperature T0

 

a = temperature coefficient of resistivity; the dimension of a is (Temperature)-1

According to the above equation, a graph of ρT plotted against T would be a straight line i.e., the resistivity of a metallic conductor increases with increasing temperature.

Different materials have a different dependence on temperature. For example, materials like Nichrome, Manganin, and constantan are less likely to change their resistivities with temperature. Hence, they are employed in wire-bound standard resistors. However, semiconductors exhibit an indirect relation with temperature. Resistivities of semiconductors decrease with increasing temperatures.

Internal Resistance:

Internal resistance is the resistance within a battery, or other voltage sources that causes a drop in the source voltage when there is a current.

The relationship between internal resistance (r) and emf (e) of cell s given by.

e = I(r + R)

Where, e = EMF i.e. electromotive force (Volts), I = current (A), R = Load resistance, and r is the internal resistance of cell measured in ohms.

On rearranging the above equation we get;

e = IR + Ir 

or, e = V + Ir

In the above equation, V is the potential difference (terminal) across the cell when the current (I) is flowing through the circuit.

Note: The emf (e) of a cell is always greater than the potential difference (terminal) across the cell.

Example: 1 The potential difference across the cell when no current flows through the circuit is 3 V. When the current I = 0.37 Ampere is flowing, the terminal potential difference falls to 2.8 Volts. Determine the internal resistance (r) of the cell?

Solution:   e = V + Ir

                    Or, (e – V)/I = r

                    r = (3.0 – 2.8)/0.37 = 0.54 Ohm.

 

Resistors in Series

Two or more resistors are said to be connected in series when the same amount of current flows through all the resistors. In such circuits, the voltage across each resistor is different. In a series connection, if any resistor is broken or a fault occurs, then the entire circuit is turned off. The construction of a series circuit is simpler compared to a parallel circuit.

For the above circuit, the total resistance is given as:

Rtotal = R1 + R2 + ….. + Rn

The total resistance of the system is just the total sum of individual resistances.

Resistors in Parallel

Two or more resistors are said to be connected in parallel when the voltage is the same across all the resistors. In such circuits, the current is branched out and recombined when branches meet at a common point. A resistor or any other component can be connected or disconnected easily without affecting other elements in a parallel circuit.

In the figure above shows the ‘n’ number of resistors connected in parallel.

The sum of reciprocals of resistance of an individual resistor is the total reciprocal resistance of the system.

Grouping of Cells:

Cells Connected in a Series:

The cells are said to be connected in series if the positive terminal of the first cell is connected to the negative terminal of the second cell, and the negative terminal of the second cell is connected to the positive terminal of the third cell. The same current flows through each cell.

Derivation:

Let us consider that ‘n’ identical cells are connected in series with the same polarity. The EMF of individual cells is E1, E2, E3 —– En. Similarly, the internal resistance of each cell is r1, r2, r3 ——–rn.

The equivalent EMF is the terminal voltage across the cell when the cell is not in use.

The equivalent EMF of the cell is given by

Eeq = E1 + E2 +E3 ——-En = nE

The equivalent internal resistance is given by

req = r1 + r2 + r3 ——-rn = nr

The combination can be replaced with a single cell of equivalent EMF ‘nE’ and equivalent internal resistance ‘nr’.

Now, the equivalent resistance of the circuit is

Req = nr + R

The current flowing through the load will be I = Eeq/Req

I = nE/(R+nr)

Case 1: If nr <<R then I = nE/R

If the value of the internal resistance is much lesser than the external resistance, then the current in the circuit will be n times the circuit current due to the single cell.

Case 2: If nr >> R then I = E/r

If the value of the internal resistance is much greater than the external resistance, then the current in the circuit will be equal to the short-circuited current obtained from a single cell.

Cells are connected in Parallel:

Parallel combination circuits have multiple paths between the terminals. In a parallel combination of cells, all the positive terminals of the cells are connected together, and the negative terminals of the cells are connected together.

Derivation:

The emf of cell 1 is ε1, and the emf of cell 2 is ε2. The internal resistance of cell 1 is r1, and cell 2 is r2. The current is split into i1 and i2. The total current i = i1 + i2

The resultant internal resistance of the combination is,

The equivalent EMF ( εeq) is equal to the potential difference between A and B (VA – VB) when it is not in use.

Apply Kirchoffs Rule for equivalent Resistance

From the figure above, we get

– ε1 + ir1 + ir2 + ε2 = 0

⇒ i = ε1– ε2/(r1 + r2) ——(1)

The potential difference VA – VB = ε1– ir1

Or VA – VB = ε2+ ir2

Substituting the value of ‘i’ in either of the two equations above, we get

VA – VB = ε2+ ir2

= (ε2r1+ ε1r2)/(r1 + r2)

Considering the equation of resultant internal resistance, the above expression can be written as

Electrical Energy and Power:

Electrical Energy:

Electrical energy is the energy derived from electric potential energy or kinetic energy of the charged particles.

                                    Or

 The electrical energy as the energy generated by the movement of electrons from one point to another. The movement of charged particles along/through a medium (say wire) constitute current or electricity

Electric Energy Formula:

let us consider a conductor carrying the current I and potential difference V between the two endpoints A and B. Let us denoted the electric potential of A and B as V(A) and V(B). As we know that current is flowing from A to B so  V(A) >V(B) and the potential difference across AB is V = V(A) – V(B) > 0

 Now, in a time interval Δt, an amount of charge ΔQ is equal to IΔt moves from point A to B of the circuit and 

W= V*ΔQ

Here if the charges in the conductor move without collisions, their kinetic energy would also change. Conservation of total energy is ΔK = I V Δt > 0. The amount of energy dissipated as heat in a conductor in a time interval Δt is,

ΔW = V ΔQ = VI Δt

Units of Electrical Energy:

*The basic unit of electrical energy is the joule or watt-second.

*The commercial unit of electrical energy is the kilowatt-hour (kWh).

1 kwh = 1000 × 60 × 60 watt – second

1 kwh = 36 × 105 Ws or Joules

 Example:-

«  A thunderstorm, is an example of electrical energy – what lightning is electricity discharging in the atmosphere.

«   Electric wheels generate electrical energy.

Power:

It is the rate at which work is done or energy is transformed in an electrical circuit.

Electrical power is denoted by P and measured using Watt

 Derivation:

The energy dissipated in time interval ∆t is given by

∆W = I V∆t

The energy dissipated per unit time is actually the power dissipated, which is given by P = ∆W/∆t

According to Ohm’s law,

V = IR

Substituting we have,

P =I2R

Or

P = V2R

It is this power which is responsible for heating up the coil of a bulb, which gives out heat and light.