Question

Dimensional formula for electromotive force (emf) (a) \(\mathrm{M}^{2} \mathrm{~L}^{1} \mathrm{~T}^{-1} \mathrm{~A}^{-3}\) (b) \(\mathrm{M}^{1} \mathrm{~L}^{2} \mathrm{~T}^{-3} \mathrm{~A}^{-1}\) (c) \(\mathrm{M}^{1} \mathrm{~L}^{1} \mathrm{~T}^{-3} \mathrm{~A}^{-1}\) (d) \(\mathrm{M}^{1} \mathrm{~L}^{2} \mathrm{~T}^{3} \mathrm{~A}^{-1}\)

Short answer

The short answer for the dimensional formula of electromotive force (emf) is \( \mathrm{M}^{1} \mathrm{~L}^{2} \mathrm{~T}^{-3} \mathrm{~A}^{-1} \).

Step-by-step solution

Write the expressions for electric current and resistance in terms of their SI units

Let's first write the dimensional formulas for electric current (I) and electrical resistance (R) in terms of their SI units. Electric current is measured in Amperes (A) and electrical resistance is measured in Ohms (Ω). Electric current (I) has a dimensional formula: [I] = A Electrical resistance (R) has a dimensional formula: [R] = V/A, where V is the Voltage which has the dimensional formula of M^(1)L^(2)T^(-3)A^(-1).

Apply Ohm's Law

Now let's apply Ohm's Law to find the dimensional formula for emf. According to Ohm's Law, emf (E) = current (I) × resistance (R). So, [E] = [I] × [R]

Substitute the Dimensional Formulae

Next, we substitute the dimensional formulae of electric current and resistance that we found in step 1 into the equation. [E] = A × (M^(1)L^(2)T^(-3)A^(-1))

Simplify the Expression

Now let's simplify this expression by combining the terms with similar dimensions. [E] = M^(1)L^(2)T^(-3)A^(1-1) [E] = M^(1)L^(2)T^(-3)A^(-1)

Identify the Correct Option

Finally, we compare the dimensional formula that we found in step 4 with the given options in the question. The correct option is (b) M^(1)L^(2)T^(-3)A^(-1).

Key concepts

Electromotive Force

Electromotive force, commonly referred to as emf, is the potential difference produced by a source of electrical energy such as a battery or generator. It is not a force in the conventional sense but a measure of energy provided per charge. When you connect a circuit, the emf is responsible for pushing the charge around the loop.
A critical aspect of understanding emf is recognizing that it is often confused with voltage. However, while voltage refers to the potential difference between two points, emf denotes the source energy that creates this difference.
Understanding emf requires knowledge of its dimensional formula. This provides insight into the physical quantities that define emf. In the SI system, the dimensional formula for emf can be derived using the units of electric current (A) and electrical resistance (Ω), resulting in:

• The formula for emf: [M^{1}L^{2}T^{-3}A^{-1}]

This formula shows that emf combines dimensions of mass (M), length (L), time (T), and current (A) in a specific manner to convey its purpose as a source of energy in electrical circuits.

Ohm's Law

Ohm's Law is a foundational concept in electronics and physics, establishing the relationship between voltage (V), current (I), and resistance (R). The law is typically expressed as:

• V = IR

This simple equation tells us that the voltage across a circuit is directly proportional to the current flowing through it, provided the resistance remains constant.
In the context of dimensional analysis, Ohm’s Law can be used to derive the dimensional formula of emf. By understanding that emf is essentially a measure of potential difference according to Ohm’s Law, we can write:

• E (emf) = I (current) × R (resistance)

This relationship allows you to view emf in terms of current and resistance, providing a basis for calculating its dimensions. Understanding this connection ensures you grasp how changes in one quantity can affect the others within an electrical system.

Dimensional Formula

The concept of dimensional formula involves representing physical quantities in terms of base dimensions such as mass (M), length (L), time (T), and electric current (A). This representation helps in checking the consistency of physical equations and converting between units.
For electromotive force (emf), the dimensional formula is derived from its basic definition in terms of voltage, which itself is derived from the physical principles demonstrated by Ohm's Law. The dimensional formula for emf, as refined in calculations, is:

• [M^{1}L^{2}T^{-3}A^{-1}]

Here, each symbol in the formula represents:

• M: Mass, relevant in mechanical interpretations of electric circuits
• L: Length, integral in determining electrical pathways
• T: Time, indicative of changes in circuit dynamics
• A: Current, the essential flow of charge defining electrical phenomena

Understanding the dimensional formula is crucial for analyzing and solving physics problems accurately. It acts as a tool that gives insight into how different quantities contribute to the behavior of electrical components.