JEE PYQ: Units & Measurements - Question ID fabae713a464 (JEE Main 2019)

Core Formula & Concept:

In dimensional analysis, every physical quantity can be expressed in terms of the fundamental dimensions: mass ($M$), length ($L$), time ($T$), and electric current ($A$). The given formula is: $X = 5 Y Z^2$ Here:

• $X$ has the dimensions of capacitance.
• $Z$ has the dimensions of magnetic field.
• $Y$ is the unknown whose dimensions we must determine.

The key formulas we use are:

1. Dimensions of capacitance ($C$): $[C] = [M^{-1} L^{-2} T^{4} A^{2}]$ This comes from the definition of capacitance in terms of charge ($Q$) and potential difference ($V$), where $Q = CV$. Charge has dimensions $[A T]$, and potential difference (voltage) has dimensions $[M L^{2} T^{-3} A^{-1}]$.
2. Dimensions of magnetic field ($B$): $[B] = [M T^{-2} A^{-1}]$ This is derived from the Lorentz force law, $F = q(v \times B)$, where force has dimensions $[M L T^{-2}]$, charge $[A T]$, and velocity $[L T^{-1}]$.

--- Step-by-Step Derivation:

We start with the given formula: $X = 5 Y Z^2$ Since the constant $5$ is dimensionless, we can ignore it for dimensional analysis. Thus: $[X] = [Y] [Z]^2$ We need to solve for $[Y]$: $[Y] = \frac{[X]}{[Z]^2}$ Step 1: Substitute dimensions of $X$ (capacitance) and $Z$ (magnetic field). $[X] = [M^{-1} L^{-2} T^{4} A^{2}]$ $[Z] = [M T^{-2} A^{-1}]$ Step 2: Compute $[Z]^2$. $[Z]^2 = \left( [M T^{-2} A^{-1}] \right)^2 = [M^2 T^{-4} A^{-2}]$ Step 3: Substitute into the equation for $[Y]$. $[Y] = \frac{[M^{-1} L^{-2} T^{4} A^{2}]}{[M^2 T^{-4} A^{-2}]}$ Step 4: Simplify the expression using the laws of exponents. When dividing dimensions, subtract the exponents of like terms:

• For $M$: $-1 - 2 = -3$
• For $L$: $-2 - 0 = -2$ (since $L$ is absent in $[Z]^2$)
• For $T$: $4 - (-4) = 8$
• For $A$: $2 - (-2) = 4$

Thus: $[Y] = [M^{-3} L^{-2} T^{8} A^{4}]$ Step 5: Match with the given options. The derived dimensions $[M^{-3} L^{-2} T^{8} A^{4}]$ correspond to Option A.

--- Common Traps & Exam Tip:

Students often make the following mistakes in this question:

1. Incorrect dimensions for capacitance or magnetic field: Some confuse the dimensions of capacitance with those of charge or electric field. Always recall that capacitance is $[M^{-1} L^{-2} T^{4} A^{2}]$, derived from $C = Q/V$. Similarly, magnetic field is $[M T^{-2} A^{-1}]$, not $[M L T^{-2} A^{-1}]$ (which is force per unit charge).
2. Forgetting to square the dimensions of $Z$: Since $Z^2$ appears in the formula, students sometimes forget to square its dimensions, leading to incorrect exponents for $T$ and $A$.
3. Sign errors in exponent subtraction: When simplifying $\frac{[X]}{[Z]^2}$, students may incorrectly add exponents instead of subtracting them. For example, $T^{4} / T^{-4} = T^{8}$, not $T^{0}$.
4. Miscounting the exponents of $L$: Since $L$ is absent in $[Z]$, its exponent in $[Y]$ should remain $-2$ (from $[X]$). Some students mistakenly change it to $0$ or another value.

Exam Tip: Always write down the dimensions of all given quantities explicitly before substituting. Double-check the exponents at each step, especially when dealing with division or powers. If time permits, verify the final dimensions by plugging them back into the original formula to ensure consistency.