Question

Which is the dimensional formula for conductance from the give below? (A) \(\mathrm{M}^{1} \mathrm{~L}^{2} \mathrm{~T}^{-3} \mathrm{~A}^{2}\) (B) \(\mathrm{M}^{-1} \mathrm{~L}^{-2} \mathrm{~T}^{3} \mathrm{~A}^{2}\) (C) \(\mathrm{M}^{1} \mathrm{~L}^{-3} \mathrm{~T}^{-3} \mathrm{~A}^{-2}\) (D) \(\mathrm{M}^{1} \mathrm{~L}^{-3} \mathrm{~T}^{3} \mathrm{~A}^{2}\)

Short answer

The short answer is: (B) \(M^{-1} L^{-2} T^{3} A^{2}\).

Step-by-step solution

Write down the formula for resistance

The formula for resistance is given by Ohm's Law, which states that: Resistance (R) = Voltage (V) / Current (I)

Write down the dimensional formulae for voltage and current

Voltage (V) has the dimension of work done per unit charge. The dimensional formula for work is [M^1 L^2 T^{-2}], and the dimensional formula for charge is [A^1 T^1]. Therefore, the dimensional formula for voltage can be represented as [M^1 L^2 T^{-3} A^{-1}]. Current (I) has the dimension of Ampere (A), so the dimensional formula for current is [A^1].

Determine the dimensional formula for resistance

Since Resistance (R) = Voltage (V) / Current (I), we find the dimensional formula for resistance by dividing the dimensional formula of voltage by the dimensional formula of current. Which gives us: [\(R = \frac{[M^1 L^2 T^{-3} A^{-1}]}{[A^1]} = M^1 L^2 T^{-3} A^{-2}\)]

Determine the dimensional formula for conductance

Conductance (G) is the reciprocal of resistance (R). So, to find the dimensional formula for conductance, we invert the dimensional formula for resistance: \[G = \frac{1}{M^1 L^2 T^{-3} A^{-2}} = M^{-1} L^{-2} T^{3} A^{2}\] The correct answer is (B) \(M^{-1} L^{-2} T^{3} A^{2}\).

Key concepts

Conductance

Conductance is a measure of how easily electricity flows through a component or a circuit. It is the inverse of resistance. While resistance measures opposition to the flow of current, conductance measures how efficiently current can pass. This concept is important in understanding various electrical devices and circuits.
A high conductance means the material allows electric current to flow easily, like a wide-open gate allowing people to pass through without difficulty. Conductance is denoted by the symbol 'G' and is quantified in Siemens (S).
Its relationship with resistance is given by the formula:

• G = 1/R

If you know the resistance of a component, you can easily find its conductance and vice versa. Understanding conductance allows engineers and scientists to design more efficient circuits and energy systems.

Ohm's Law

Ohm's Law is a fundamental principle in the study of electricity. It relates voltage (V), current (I), and resistance (R) in an electrical circuit. The law is both simple and powerful:

• V = IR

This formula tells us that the voltage across a conductor is proportional to the current flowing through it, multiplied by the resistance. It is a key point in solving many electrical problems.
To understand this, think of it like water flowing through a pipe. The voltage is the pressure pushing the water, the current is the flow rate of the water, and the resistance is like the width of the pipe. The broader the pipe (lower resistance), the easier the flow (higher current) for a given pressure (voltage).
Knowing Ohm's Law is crucial for calculating any missing parameters when dealing with electrical circuits and helps in designing electronic systems.

Dimensional Formula

A dimensional formula expresses the dependence of a physical quantity on the base quantities like mass, length, time, and electric current. It provides insight into the nature of the quantity, ensuring consistency in equations.
For example, if we consider voltage, its dimensional formula is

• [M^1 L^2 T^{-3} A^{-1}]

This tells us that voltage is composed of mass, length, time, and electric current dimensions in specific powers. Calculating the dimensional formula helps in deriving relationships, checking equations, and ensuring units are consistent.
In this context, understanding the dimensional formula of conductance helps in solving various physics and engineering problems by providing a clearer view of how different physical quantities relate to each other.

Resistance

Resistance is a fundamental concept in electricity, defined by its opposition to the flow of electric current. It's like trying to drive on a busy highway where the traffic slows you down. Resistance is measured in ohms (Ω) and is affected by factors like the material, length, and thickness of the conductor.
In essence, materials with high resistance don't allow current to flow easily, while low resistance materials do. The resistor is a basic electronic component designed to introduce resistance into a circuit, intentionally reducing current flow.

• Metal wires usually have low resistance, which is why they are often used in electrical circuits.
• Materials like rubber have high resistance, acting as insulators.

Thus, understanding resistance not only allows for controlling current flows in circuits but is also essential when selecting materials for wiring and insulation in various applications.