Electrical Resistance and Resistivity Test - 1

Explanation:

• Internal resistance (r): The resistance offered by the electrolyte to the flow of ions is known as the internal resistance.

A cell is said to be ideal if it has zero internal resistance.

CONCEPT:

Resistance: The obstruction offered to the flow of current is known as the resistance. It is denoted by R.

• When two or more resistances are connected one after another such that the same current flows through them then it is called resistances in series.

The equivalent resistance in series combination is will be

R_ser = R₁ + R₂ 

When the terminals of two or more resistances are connected at the same two points and the potential difference across them is equal then it is called resistances in parallel.

The net resistance/equivalent resistance(R) of resistances in parallel is given by:

\(\frac{1}{R} = \frac{1}{{{R_1}}} + \frac{1}{{{R_2}}}\)

CALCULATION:

In the above figure, 6Ω and 6Ω are connected in parallel.

The net resistance/equivalent resistance(R) of two 6Ω resistances connected in parallel is given by:

\(\frac{1}{{R'}} = \frac{1}{{{R_1}}} + \frac{1}{{{R_2}}}\)

\(\Rightarrow \frac{1}{{R'}} = \frac{1}{6} + \frac{1}{6} = \frac{2}{6} = \frac{1}{3}\)

R’ = 3 Ω

Now R’ and 3 Ω are in series,

∴ R_net = R’ + 3 Ω

⇒ R_net = 3 Ω + 3 Ω = 6 Ω

Parallel grouping:

In parallel grouping, all anodes are connected at one point and all cathode are connected together at other points.

If n identical cells are connected in parallel

1. Equivalent emf of the combination Eeq = E
2. Equivalent internal resistance req = r/n

As we know that according to ohm's law,

V = IR

Where R = resistance, I = current and V = electric potential 

\(\Rightarrow I = \frac{V}{R}\)

Therefore when cells are connected in parallel, the voltage remains the same, and internal resistance decreases, hence the current increases. Therefore option 3 is correct.

Important Points

Serial grouping:

In a series grouping of cell’s their emf’s are additive or subtractive while their internal resistances are always additive.

CONCEPT:

• Grouping of Cells: In a series grouping of cell’s their emf’s are additive or subtractive while their internal resistances are always additive.
• If dissimilar plates of cells are connected together then their emf’s are added to each other while if their similar plates are connected together then their emf’s are subtractive

Series grouping: 

• In series grouping anode of one cell is connected to the cathode of other cells and so on.

• If n identical cells are connected in series

1. Equivalent emf of the combination Eeq = nE
2. Equivalent internal resistance req = nr

Parallel grouping:

• In parallel grouping, all anodes are connected at one point and all cathode are connected together at other points.

• If n identical cells are connected in parallel

1. Equivalent emf of the combination Eeq = E
2. Equivalent internal resistance req = r/n

CALCULATION:

Given - number of cell = 5 and internal resistances (r) = 1.5 Ω

• The equivalent internal resistance of the cells is 

⇒ req = r/n

⇒ req = 1.5/5 = 0.3 Ω 

Concept:

The below equation shows the method to find the resistance:

Starting from the right, the first band gives the tolerance value of the resistor.

Resistor Colour Coding uses colored bands to easily identify a resistor resistive value and its percentage tolerance. The resistor color code markings are always read one band at a time starting from the left to the right, with the larger width tolerance band oriented to the right side indicating its tolerance. 

The value of the resistance is given in the form:

R = AB × C ± D%

Where,

‘A’ and ‘B’ indicate the first two significant figures of resistance (Ohms).

‘C’ indicates the decimal multiplies.

‘D’ indicates the tolerance in percentage.

The table for the resistor color code is given below:

Calculation:

From the above resistance color codes,

Blue → 6

Grey → 8

Orange → 103

Silver → ± 10 %

Now, the resistance is:

R = 68 × 103 Ω ± 10 %

R = 68000 ± 10% ohms

Shortcut Trick

BB ROY Great Britain Very Good Watch Gold and Silver

B - Black (0)

B - Brown (1)

R - Red (2)

O - Orange (3)

Y - Yellow (4)

G - Green (5)

B - Blue (6)

V - Violet (7)

G - Grey (8)

W - Whilte (9)

Tolerance - Gold(5%) and Silver(10%)

CONCEPT:

• Resistance: The property of electrical materials and electrical instruments that opposes the flow of current through them is called resistance.
  • It is denoted by R. The SI unit of resistance is the ohm (Ω).
• Power: The rate of work done by an electric current is called power. It is denoted by P. The SI unit of power is the watt (W).

Power dissipation is given by:

Power (P) = V I = V2/R = I2 R

Where V is the potential difference across resistance, I is current flowing and R is resistance.

CALCULATION:

Given that: Power rating of bulb = (10 W, 220 V)

Power (P) of bulb at 220 V = 10 W

Power (P) = V2/R 

Resistance (R) = V²/P = 220²/10 = 4840 Ω

So option 3 is correct.

CONCEPT:

• Resistance: The obstruction offered to the flow of current is known as the resistance. It is denoted by R.
• When two or more resistances are connected one after another such that the same current flows through them then it is called resistances in series.
• The equivalent resistance in series combination is will be

⇒ Rser = R1 + R2 + R3

• When the terminals of two or more resistances are connected at the same two points and the potential difference across them is equal then it is called resistances in parallel.

• The net resistance/equivalent resistance(R) of resistances in parallel is given by:

\(⇒ \frac{1}{R} = \frac{1}{{{R_1}}} + \frac{1}{{{R_2}}}\)

EXPLANATION:

Given that,

R1 = R2 = R3 = R

• When the resistor is connected in parallel, then the equivalent resistance is

\(⇒ \frac{1}{{{R_{para}}}} = \frac{1}{{{R_1}}} + \frac{1}{{{R_2}}} + \frac{1}{{{R_2}}} = \frac{1}{R} + \frac{1}{R} + \frac{1}{R} = \frac{3}{R}\)

⇒  Rpara = R/3 

• Therefore, the total resistance in the circuit becomes one-third of the individual resistance.

CONCEPT:

• The rate of work done by the electric current is called as electric power.
• The difference in potential between two points is called a potential difference.

Electric energy (E) = electric power (P) × time (t)

1 unit of energy = 1 KWh

EXPLANATION:

• The commercial unit of electric energy is kilowatt or the Board of Trade (B.O.T) unit.
• One kilowatt-hour is defined as the electric energy consumed by the appliance of 1 kilowatt in one hour and it is also called one unit of energy.

1 kilowatt-hour = 1 kilowatt × 1 hours = 1000 watt × 3600 s

CONCEPT:

• Resistivity (ρ): The property of a conductor that opposes the flow of electric current through them and independent of the shape and size of the materials but depends on the nature and temperature of the materials is called resistivity.
  • The unit for resistivity is the ohm-meter (Ω-m).
  • The resistivity of a material depends on its nature and the temperature of the conductor.
  • The resistivity of a material doesn't depend on its shape and size (length and area).
  • Materials that conduct electrical current easily are called conductors and have a low resistivity.
  • Materials that do not conduct electricity easily are called insulators and these materials have a high resistivity.
  • Resistivity is inversely proportional to the number of free electrons per unit volume of the conductor and to the average relaxation time of the free electrons in the conductor.
• Resistance: The property of any conductor that opposes the flow of electric current through it and depends on the shape and size of the materials, temperature, and nature of the materials is called resistance.
  • It is denoted by R and the SI unit is the ohm (Ω).

The resistance is given by:

R = ρL/A

where ρ is resistivity, L is the length and A is the area of the cross-section. 

EXPLANATION:

From the above discussion, we can say that

1. The resistivity doesn't depend on the dimensions (length, diameter, and area) of the conductor. 
2. The resistivity depends on the material of the conductor. So option 4 is correct.
3. The resistivity depends on the temperature of the conductor. But the effect of temperature is negligible.
4. The resistivity does not depend on the density of the conductor.

EXTRA POINTS:

Difference between resistivity and resistance:

CONCEPT:

Thermometer:

• An instrument used to measure the temperature of a body is called a thermometer.
• Resistance thermometers: Usually platinum is used in resistance thermometers due to the high melting point and large value of temperature coefficient of resistance.
• For the resistance thermometer,

\(⇒ t=\frac{R-R_o}{R_{100}-R_o}\times100^o C\)

Where R = resistance at temperature t, R_o = resistance at temperature 0° C, and R₁₀₀ = resistance at temperature 100° C

CALCULATION:

Given Ro = 5.8Ω, R100 = 8.4Ω, and R = 6.2Ω

• For the resistance thermometer,

\(⇒ t=\frac{R-R_o}{R_{100}-R_o}\times100^o C\)

\(⇒ t=\frac{6.2-5.8}{8.4-5.8}\times100^o C\)

⇒ t = 15.4° C

• Hence, option 3 is correct.