Resistance and Resistivity MCQ Quiz - Objective Question with Answer for Resistance and Resistivity - Download Free PDF

Key Points

• The resistance (R) of a wire is given by the formula R = ρ(L/A), where ρ is the resistivity, L is the length, and A is the cross-sectional area.
• Since both wires have the same length and resistance, we can set up a ratio for their resistivities based on their cross-sectional areas.
• The ratio of their cross-sectional areas is 1 : 8, implying one wire has a cross-sectional area 8 times larger than the other.
• Given that resistance is inversely proportional to the area (A), the wire with the larger area will have a lower resistivity if all other factors are constant. However, since their resistances are the same and lengths are equal, the only way to balance the equation is if their resistivities are directly proportional to their areas, resulting in the ratio of resistivities being  ρ = R(A/L)
• \(\rho \propto \text{Area} \\ A_1:A_2=1:8 \\ \implies \rho_1:\rho_2=1:8 \ \ \)

Additional Information

Resistivity is a fundamental property of materials that quantifies how strongly a material opposes the flow of electric current. A low resistivity indicates a material that readily allows the flow of electric current. Resistivity is specific to each material and influences how it is used in electrical and electronic applications. For example:

• Copper has a low resistivity, making it an excellent choice for electrical wiring.
• Rubber, on the other hand, has a high resistivity, which makes it an effective insulator.
• The resistivity of materials can be affected by temperature; in most metals, resistivity increases with temperature.
• Semiconductors have resistivities that fall between those of conductors and insulators, which can be altered by doping with impurities and changes in temperature.

\(R = P \dfrac{L}{A}\)

For the same wire, \(P\) is constant.

\(R_1 = P \dfrac{L_1}{A_1}, \quad R_2 = P \dfrac{L_2}{A_2}\)

\(\dfrac{R_1}{R_2} = \dfrac{L_1}{A_1} \times \dfrac{A_2}{L_2} \quad \longrightarrow (i)\)

As the same wire is stretched, so volume remains constant, i.e.,

\(L_1 A_1 = L_2 A_2 \quad \longrightarrow (ii)\)

From (i) and (ii):

\(\dfrac{R_1}{R_2} = \left( \dfrac{A_2}{A_1} \right)^2\)

\(\dfrac{R_1}{R_2} = \left( \dfrac{d_2^2}{d_1^2} \right)^2\)

\(\Rightarrow \boxed{\dfrac{R_1}{R_2} = \dfrac{d_2^4}{d_1^4}}\)

Concept:

Shunt Resistance:

• A shunt resistor is used to bypass a portion of current around a device, typically in ammeters.
• The shunt should have very low resistance to prevent significant voltage drop and to allow most of the current to pass through it.
• A low resistance is achieved by making the shunt wire short (to reduce length) and thick (to allow more current without increasing resistance).
• A thick wire reduces the resistance as it offers less opposition to current flow compared to a thinner wire.

Explanation:

Given the options:

• 1) Short and thin: This would have higher resistance, not ideal for a shunt.
• 2) Long and thin: This would have even higher resistance, not ideal for a shunt.
• 3) Long and thick: Though the thickness is good, the long length would increase resistance, so not ideal.
• 4) Short and thick: This is ideal as it minimizes resistance and allows the current to bypass effectively.

∴ The correct answer is option 4: short and thick.

The correct answer is Ω .

Key Points

• The ohm (symbol: Ω) is the SI derived unit of electrical resistance.
• It is named after German physicist Georg Simon Ohm.
• An ohm is defined as the electrical resistance between two points of a conductor when a constant potential difference of one volt, applied to these points, produces a current of one ampere.
• The unit is represented by the Greek letter Omega (Ω).
• The concept is essential in understanding and analyzing electrical circuits.

Additional Information

• Georg Simon Ohm
  • He was a German physicist and mathematician.
  • Ohm is best known for Ohm's Law, which states that the current through a conductor between two points is directly proportional to the voltage across the two points.
• Ohm's Law
  • The formula is V = IR, where V is voltage, I is current, and R is resistance.
  • It is a fundamental principle used in electrical engineering and physics.
• SI Units
  • SI stands for the International System of Units.
  • It is the modern form of the metric system and the world's most widely used system of measurement.
• Electrical Resistance
  • It is a measure of the difficulty to pass an electric current through a conductor.
  • Materials with low resistance are good conductors, while materials with high resistance are good insulators.

CONCEPT:

Resistivity (ρ):

• Resistivity is numerically equal to the resistance of a substance having a unit area of cross-section and unit length.
• The formula of resistivity is \(\rho = \frac{m}{{n{e^2}\tau }}\)

Where, m = mass of an electron, n = Number of electron per unit volume of the conductor, τ = relaxation time.

EXPLANATION:

• From the above equation, it is clear that resistivity of the conductor depends on the number of electrons per unit volume of the conductor. Therefore option 1 is incorrect.
• From the above equation, it clear that resistivity depends on relaxation time (τ).

As we know τ ∝ 1/T

• ρ ∝ T. Hence, resistivity increases with an increase in temperature and vice-versa. Therefore option 2 is correct.
• Resistivity is the intrinsic property of the substance. It is independent of the shape and size of the body i.e. on length and area. Therefore option 3 is incorrect.

Mistake Points

• SI unit of conductance is ohm-1 (mho) or ohm per meter or seimens.
• Here given ohm meter which is wrong

CONCEPT:

• Electrical conductance (K): The reciprocal of electrical resistance is called conductance.
  •  It is a measure of how easily current can flow through a material.
  • Its SI unit is Siemens.

Electrical conductance (K) = 1/R 

• Electrical resistance (R): The electrical resistance of a material refers to its opposition to the flow of current.
  • Its SI unit is Ohm (Ω). 

EXPLANATION:

The SI unit of conductance = 1/ (The SI unit of resistance) = 1/Ω = Ω^-1 = Siemens. Hence option 2 is correct.

• Ohm is the SI unit of resistance.
• Ampere is the SI unit of current.
• Ohm-meter is the SI unit of electrical resistivity which is the measure of an object's ability to oppose the flow of current.

CONCEPT:

• Electrical Conductivity: It is a property of a material that shows the ability of a material to carry current.
  • It is the opposite of electrical resistivity.
• The below table shows the electrical conductivity of different materials:

EXPLANATION:

• From the given options, Silver has the highest electrical conductivity, and Aluminum has the least.
• On a scale of 0 to 100 for electrical conductivity, silver ranks 100, with copper at 97 and gold at 76, aluminum at 60.
• Hence the correct answer is option 4.

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: 

• Resistance: The property of any conductor that opposes the flow of electric current through it 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 given- formula of the resistance:

• The resistance of a uniform metallic conductor is inversely proportional to its area and directly proportional to length. So option 3 is correct.
• ρ depends upon the material and the temperature.
• The resistance of a uniform metallic conductor is dependent on length, area, and temperature also.

CONCEPT:

• Resistivity: The measure of the resistance of a specific material to electrical conduction is called resistivity. It is denoted by ρ. 
  • Resistivity is the property of any material/conductor.
  • It also depends on temperature and nature but not on shape and size.
  • Its unit is ohm-meter.

EXPLANATION:

• Since the resistivity of any conductor is the property of the material. So it is independent of the shape and the size. 
• It depends on the material of the conductor. So option 1 is correct.
• Electrical resistivity (ρ)
  • It is defined as the resistance of a unit length with a unit area of cross-section of the material of the conductor.

​ 

• Difference between resistivity and resistance:

Concept: 

Resistance: 

• The measurement of the opposition of the flow of electric current through a conductor is called the resistance of that conductor. It is denoted by R.

There are mainly two ways of the combination of resistances:

1. Resistances in series:

• When two or more resistances are connected one after another such that the same current flows through them are called as resistances in series.
• The net resistance/equivalent resistance (R) of resistances in series is given by:
• Equivalent resistance, R = R1 + R2

2. Resistances in parallel:

• When the terminals of two or more resistances are connected at the same two points and the potential difference across them is equal is called resistances in parallel.
• The net resistance/equivalent resistance(R) of resistances in parallel is given by:

\(\frac{1}{{{R_{eff}}}} = \frac{1}{{{R_1}}} + \frac{1}{{{R_2}}}\)

Explanation:

When three resistors of 3 Ω are connected in parallel.

\(\frac{1}{{{R_{eff}}}} = \frac{1}{{{R_1}}} + \frac{1}{{{R_2}}} + \frac{1}{{{R_3}}}\)

\(\frac{1}{{{R_{eff}}}} = \frac{1}{{{3}}} + \frac{1}{{{3}}} + \frac{1}{{{3}}}\)

R_eff = 1Ω 

Additional Information

CONCEPT:

• An electrical safety device that is used to protect the electrical circuit from the overloads is called fuse.

The material used for fuse elements must be of:

• low melting point, low ohmic loss,
• high resistivity,
• low cost and free from detraction.
• The material used for making fuse element has a low melting point such as tin, lead, or zinc.
• A low melting point is, however, available with a high specific resistance metal.
• The material mainly used for fuse element are tin, lead, silver, copper, zinc, aluminum,  and an alloy of lead and tin

EXPLANATION:

• Hence from the given explanation above, an electrical fuse must have high resistivity to avoid loss of current and low melting point so that in case of short-circuiting. Therefore option 4 is correct.

• The current rating of the fuse is the maximum value of current due to which fuse does not get melt.
• The current carrying capacity of a fuse wire depends upon
  • The material used for it
  • The dimensions of the fuse i.e. diameter and length, size and shape of terminals used to connect it
  • Surrounding of fuse

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.
• 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:

• The resistance offered by a wire of unit length and unit area of cross-section is called resistivity or specific resistance (r).​

\(ρ=R.\frac{A}{L}\)

• The SI unit of length (L) is m and area (A) is m².

Putting on the respective units, we get:

\(ρ=Ohm\times \frac{metre^2}{metre}\)

The SI unit of ρ = Ohm-meter (Ω-m).

• Thus the SI unit of resistivity is Ω-m. So option 1 is correct.

CONCEPT:

• The resistance of a wire with length l and cross-sectional area A is given as 

\(R = ρ \frac{l}{A}\)

ρ is the resistivity of the material of resistance which depends upon material and temperature and not on dimensions of wire.

• The volume of wire is given as the product of its length (l) and the cross-sectional area (A)  is given as 

V = l.A

When the wire is stretched its volume remains the same.

CALCULATION:

So, we have a wire with length l and cross-sectional area A, its resistance is given as

\(R = ρ \frac{l}{A}\)  -- (1)

if the length is increased twice, the new length is l' = 2l --- (2)

Then the cross-sectional Area is also affected and the new cross-sectional area is A'

Since volume remains the same

l.A = l'.A'

⇒ l.A = 2l.A'

⇒ A' = A / 2 --- (3)

The resistance of this stretched wire is 

\(R' = ρ \frac{l'}{A'}\) --- (4)

Putting (2) and (3) in (4)  we get

\(R' = ρ \frac{2l}{A/2}\)

⇒ \(R' = 4ρ \frac{l}{A}\) = 4 R (from (1))

So, R' = 4R

Given R = 4Ω

So, R' = 4 × 4Ω = 16Ω

Hence the resistance of wire is now 16Ω.

• Aspirants forget to consider that area also changes while stretching the wire. They only consider the change in length and do the calculations.

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

• The resistivity depends on the nature of the material.
• The resistivity depends on the average relaxation time and the number of free electrons per unit volume. 
• The resistivity doesn't depend on the dimensions (length and area) of the material. So option 3 is correct.

EXTRA POINTS:

Difference between resistivity and resistance: