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Chapter 12: Problem 1836

Match the physical quantities given in column I with their dimensional formulae given in column II - I stands for the dimension of current. \(\begin{array}{ll}\text { Column I } & \text { Column II }\end{array}\) (a) Electromotive force (emf) (p) \(\mathrm{ML}^{2} \mathrm{~T}^{-3} \mathrm{~A}^{-2}\) (b) Resistance (q) \(\mathrm{ML}^{3} \mathrm{~T}^{-3} \mathrm{~A}^{-2}\) (c) Resistivity (r) \(\mathrm{M}^{-1} \mathrm{~L}^{-3} \mathrm{~T}^{3} \mathrm{~A}^{2}\) (d) Conductivity (s) \(\mathrm{ML}^{2} \mathrm{~T}^{-3} \mathrm{~A}^{-1}\) (A) \(a-s, b-p, c-q, d-r\) (B) \(a-p, b-s, c-r, d-p\) (C) \(a-p, b-s, c-r, d-q\) (D) \(a-r, b-p, c-q, d-s\)

Short Answer

Expert verified
The short answer is: (A) \(a-s, b-p, c-q, d-r\)

Step by step solution

01

Find the dimensions of Electromotive force (emf)

The electromotive force (emf) is the work done per unit charge. The dimension of work is [M鹿L虏T鈦宦瞉, and the dimension of charge is [A鹿T鹿], so the dimension of emf is: \[ \frac{[M^1 L^2 T^{-2}]}{[A^1 T^1]} = [M^1 L^2 T^{-3} A^{-1}]. \] Looking at Column II, emf matches with (s).
02

Find the dimensions of Resistance

Resistance is given by Ohm's Law: V = IR, where V is the voltage (dimensions of emf which is [M鹿L虏T鈦宦矨鈦宦筣), I is current (dimension [A鹿]), and R is the resistance. The dimension of resistance is: \[ \frac{[M^1 L^2 T^{-3} A^{-1}]}{[A^1]} = [M^1 L^2 T^{-3} A^{-2}]. \] Looking at Column II, Resistance matches with (p).
03

Find the dimensions of Resistivity

Resistivity is given by the equation R = 蟻(L/A) where R is resistance (dimensions of [M鹿L虏T鈦宦矨鈦宦瞉), L is length (dimension [L鹿]), and A is the area (dimension [L虏]). The dimension of resistivity is: \[ \frac{[M^1 L^2 T^{-3} A^{-2}]}{[L^1]} \cdot [L^2] = [M^1 L^3 T^{-3} A^{-2}]. \] Looking at Column II, Resistivity matches with (q).
04

Find the dimensions of Conductivity

Conductivity is the inverse of resistivity. So the dimensions of conductivity are the inverse of the dimensions of resistivity, which are [M鹿L鲁T鈦宦矨鈦宦瞉: \[ \frac{1}{[M^1 L^3 T^{-3} A^{-2}]} = [M^{-1} L^{-3} T^3 A^2]. \] Looking at Column II, Conductivity matches with (r).
05

Choose the correct answer

As per our analysis, we have: (a) with (s), (b) with (p), (c) with (q), and (d) with (r). The correct option is: (A) \(a-s, b-p, c-q, d-r\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Electromotive Force
Electromotive force (emf) is a fundamental concept in physics, particularly in the study of electricity. It's the force that drives an electric current through a circuit. Essentially, it's the energy provided by a source, such as a battery, to move charges around the circuit. The dimensional formula for emf is derived from the work done per unit charge:
  • Work done: \[M^1 L^2 T^{-2}\]
  • Charge: \[A^1 T^1\]
Therefore, the dimensional formula for emf is:\[\frac{[M^1 L^2 T^{-2}]}{[A^1 T^1]} = [M^1 L^2 T^{-3} A^{-1}]\]This shows that emf has dimensions of mass, length, time, and current, reflecting its role in energy transformation.
Resistance
Resistance is a measure of how much a material opposes the flow of electric current. It's crucial in determining how much current will flow for a given voltage applied according to Ohm's Law: \(V = IR\) where \(V\) is voltage, \(I\) is current, and \(R\) is resistance. The dimensional formula for resistance can be derived using the dimensions of voltage and current:
  • Voltage (emf): \[M^1 L^2 T^{-3} A^{-1}\]
  • Current: \[A^1\]
Thus, the dimensional formula for resistance is:\[\frac{[M^1 L^2 T^{-3} A^{-1}]}{[A^1]} = [M^1 L^2 T^{-3} A^{-2}]\]Resistance quantifies the difficulty electrons face in passing through a conductor.
Resistivity
Resistivity is a material-specific property that affects how much resistance is offered per unit length and cross-sectional area. It's represented by the symbol \(\rho\). The formula for resistivity is:\[R = \rho \left(\frac{L}{A}\right)\]where:
  • \(R\) is resistance \[M^1 L^2 T^{-3} A^{-2}\]
  • \(L\) is length \[L^1\]
  • \(A\) is cross-sectional area \[L^2\]
The dimensional formula for resistivity is:\[\frac{[M^1 L^2 T^{-3} A^{-2}]}{[L^1]} \cdot [L^2] = [M^1 L^3 T^{-3} A^{-2}]\]It reflects how intrinsic properties of a material affect its electrical resistance.
Conductivity
Conductivity is the measure of a material's ability to conduct electric current. It is the reciprocal of resistivity, denoted by \(\sigma\). A high conductivity indicates that a material allows electricity to flow freely. The dimensional formula for conductivity, being the inverse of resistivity, is:\[\frac{1}{[M^1 L^3 T^{-3} A^{-2}]} = [M^{-1} L^{-3} T^3 A^2]\]This formula indicates how easily electric charges can move through a material. Understanding conductivity is essential for selecting materials for specific electrical applications, like wiring or semiconductor technology.

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Most popular questions from this chapter

In the circuit shown in fig the potential difference across \(3 \Omega\) is. (A) \(2 \mathrm{~V}\) (B) \(4 \mathrm{~V}\) (C) \(8 \mathrm{~V}\) (D) \(16 \mathrm{~V}\)

What maximum power can be obtained from a battery of \(\mathrm{emf} \mathrm{E}\) and internal resistance \(\mathrm{r}\) connected with an external resistance \(\mathrm{R}\) ? (A) \(\left(E^{2} / 4 r\right)\) (B) \(\left(E^{2} / 3 r\right)\) (C) \(\left(E^{2} / 2 r\right)\) (D) \(\left(\mathrm{E}^{2} / \mathrm{r}\right)\)

Assertion and reason are given in following questions each question has four options one of them is correct select it. (a) Both assertion and reason are true and the reason is correct reclamation of the assertion. (b) Both assertion and reason are true, but reason is not correct explanation of the assertion. (c) Assertion is true, but the reason is false. (d) Both, assertion and reason are false. Assertion: the drift velocity of electrons in a metallic wire will decrease, if the temperature of the wire is increased Reason: On increasing temperature, conductivity of metallic wire decreases. (A) a (B) \(b\) (C) \(\mathrm{c}\) (D) d

An electrical circuit is shown in fig the values of resistances and the directions of the currents are shown A voltmeter of resistance \(400 \Omega\) is connected across the \(400 \Omega\) resistor. The battery has negligible internal resistance.The residing of the voltmeter is. \((\mathrm{A})(10 / 3) \mathrm{V}\) (B) \(5 \mathrm{~V}\) (C) \((20 / 3) \mathrm{V}\) (D) \(4 \mathrm{~V}\)

Three identical resistors connected in series with a battery, together dissipate \(10 \mathrm{~W}\) of power. What will be the power dissipated, if the same resistors are connected in paralle1 across the same battery? (A) \(60 \mathrm{~W}\) (B) \(30 \mathrm{~W}\) (C) \(90 \mathrm{~W}\) (D) \(120 \mathrm{~W}\)
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