Dimension of a physical quantity is defined as

General
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dimension of a physical quantity is defined as

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Dimension of a Physical Quantity is Defined As

Answer:
The dimension of a physical quantity refers to the powers to which the base quantities, like mass, length, time, electric current, temperature, amount of substance, and luminous intensity, are raised to represent that quantity. It describes the nature of the physical quantity and helps in understanding the type, consistency, and relationship between different physical units.

Detailed Explanation:

  1. Basic Dimensions (Base Quantities):

    • The seven fundamental dimensions in physics and their standard symbols are:
      • Mass ((M))
      • Length ((L))
      • Time ((T))
      • Electric Current ((I))
      • Temperature ((\Theta))
      • Amount of Substance ((N))
      • Luminous Intensity ((J))

  2. Dimensional Formula:

    • A dimensional formula expresses a physical quantity in terms of the basic dimensions. For example:

      • Velocity: The velocity ( v ) is the rate of change of displacement (length) with time. This can be written as:

        [v] = \frac{L}{T} = LT^{-1}

      • Force: The force ( F ) is defined as mass times acceleration. Acceleration is the rate of change of velocity, which is length per time squared. Therefore, the dimensional formula for force is:

        [F] = M \times \left(\frac{L}{T^2}\right) = MLT^{-2}

  3. Dimensional Analysis:

    • Homogeneity of Dimensions: In any physically meaningful equation, the dimensions on both sides must be the same. This principle is known as the dimensional homogeneity. For example, in the equation ( s = ut + \frac{1}{2}at^2 ), the dimensions of each term must be the same:

      \text{Dimension of } s = L \\ \text{Dimension of } ut = LT^{-1} \times T = L \\ \text{Dimension of } \frac{1}{2}at^2 = \left(\frac{L}{T^2}\right) \times T^2 = L

    • Checking Consistency: Dimensional analysis is often used to check the consistency of physical equations and to derive relations among physical quantities.

    • Conversion of Units: It aids in converting units from one system to another by maintaining the dimensional consistency.

Importance of Dimensional Analysis:

  1. Verification of Equations:
    • It helps in verifying the correctness of physical equations by ensuring dimensional consistency.
  2. Derivation of Relations:
    • Dimensional analysis is useful in deriving relations between different physical quantities without having a detailed understanding of the processes involved.
  3. Unit Conversion:
    • It assists in converting physical quantities from one set of units to another while ensuring that the representation remains meaningful.

Final Answer:
The dimension of a physical quantity is defined as the expression that shows how the quantity is dependent on the basic dimensions (mass, length, time, etc.) in terms of their powers. This is crucial for understanding the relationships between different physical quantities and ensuring the consistency and correctness of physical equations.