Problem 3 - Master Resource Book in JEE Main Physics

Question

In the equation \(X=3 Y Z^{2}, X\) and \(Z\) have dimensions of capacitance and magnetic induction respectively. In MKSQ system, the dimensional formula of \(Y\) is (a) \(\left[\mathrm{M}^{-3} \mathrm{~L}^{-2} \mathrm{~T}^{-2} \mathrm{Q}^{-4}\right]\) (b) \(\left[\mathrm{ML}^{-2}\right]\) (c) \(\left[\mathrm{M}^{-3} \mathrm{~L}^{-2} \mathrm{Q}^{4} \mathrm{~T}^{8}\right]\) (d) \(\left[\mathrm{M}^{-1} \mathrm{~L}^{-2} \mathrm{Q}^{4} \mathrm{~T}^{4}\right]\)

Step-by-Step Solution

Answer

(c) \( [M^{-3} L^{-2} T^8 Q^4] \)

Step 1: Understand the Given Expression

The given expression is \( X = 3YZ^2 \). Here, \( X \) has the dimension of capacitance, and \( Z \) has the dimension of magnetic induction. We are asked to find the dimensional formula of \( Y \) in the MKSQ system.

Step 2: Write Dimensional Formulas for Known Quantities

In the MKSQ system, the dimensional formula for capacitance \( C \) is \( [M^{-1} L^{-2} T^4 Q^2] \), and for magnetic induction \( B \) is \( [M^1 T^{-2} Q^{-1}] \).

Step 3: Substitute and Rearrange Equation

Substitute the dimensional formulas in the expression \( X = 3YZ^2 \). Since \( 3 \) is dimensionless, we have: \[ [M^{-1} L^{-2} T^4 Q^2] = [Y][M^1 T^{-2} Q^{-1}]^2 \] Expanding the right side gives: \[ [M^{-1} L^{-2} T^4 Q^2] = [Y][M^2 T^{-4} Q^{-2}] \]

Step 4: Solve for Dimension of Y

To solve for \( Y \), divide both sides by \( [M^2 T^{-4} Q^{-2}] \): \[ [Y] = \frac{[M^{-1} L^{-2} T^4 Q^2]}{[M^2 T^{-4} Q^{-2}]} \] Simplify the expression, we get:\[ [Y] = [M^{-1-2} L^{-2} T^{4+4} Q^{2+2}] \] This simplifies to:\[ [Y] = [M^{-3} L^{-2} T^8 Q^4] \]

Step 5: Match with Options

Compare the derived dimensional formula \( [M^{-3} L^{-2} T^8 Q^4] \) of \( Y \) with the given options. Option (c) \( [M^{-3} L^{-2} Q^4 T^8] \) matches our result.

Key Concepts

Capacitance in MKSQ System

Capacitance is an important concept in physics regarding the ability of a system to store electrical charge. In the MKSQ (Meter-Kilogram-Second-Coulomb) system, the dimensional formula for capacitance is \[ [M^{-1} L^{-2} T^4 Q^2] \]. This breaks down as follows:

• \(M^{-1}\): Involves mass, indicating that capacitance is inversely related to mass.
• \(L^{-2}\): Involves length, illustrating its inverse relation with square of length.
• \(T^4\): Involves time, suggesting a relation with the fourth power of time.
• \(Q^2\): Relates to charge, showing direct proportionality to the square of charge.

Capacitance is symbolized as \(C\) and can be thought of as the measure of how much charge can be stored per unit voltage. It is a key factor in designing and understanding electrical circuits. Understanding how capacitance dimensions relate to other physical quantities is crucial for solving problems involving electrical storage and transfer.

Magnetic Induction

Magnetic induction, often referred to as magnetic flux density, represents how a magnetic field influences its surroundings. In the MKSQ system, its dimensional formula is \[ [M^1 T^{-2} Q^{-1}] \]. This concept forms the backbone of electromagnetism and informs us about how magnetic fields interact with charged particles.

• \(M^1\): Indicates that magnetic induction is directly related to mass.
• \(T^{-2}\): Shows inverse proportionality to the square of time, highlighting how rapid changes in time can enhance magnetic effects.
• \(Q^{-1}\): The inverse relation to charge implies that as the charge increases, the magnetic induction decreases.

Magnetic induction is typically denoted by the symbol \(B\) and is measured in teslas. This concept is extensively applied in designing generators, transformers, and other devices that harness magnetic fields.

Dimensional Formula

The dimensional formula of a physical quantity expresses its dependence on the base quantities (mass, length, time, charge) of a system. It provides a framework for verifying equations for dimensional consistency and understanding the nature of physical relationships.In the original problem, the expression was given as \( X = 3YZ^2 \). By breaking down the dimensions of each part of the equation and rearranging, we can solve for the unknown dimensional formula of \(Y\).

• Step 1: Recognize given dimensions: \(X\) (capacitance) and \(Z\) (magnetic induction).
• Step 2: Express these using their dimensional formulas, \(X = [M^{-1}L^{-2}T^4Q^2]\) and \(Z = [M^1 T^{-2} Q^{-1} ]\).
• Step 3: Rearrange to isolate \([Y]\).

Dimensional analysis helps ensure that physical equations make sense and offers a way to derive or check the relationships between quantities. The final derived formula \([Y] = [M^{-3}L^{-2}T^8Q^4]\) demonstrates the critical role dimensional formulas play in physics.