JEE PYQ: Units & Measurements - Question ID 8fb1bc7570c8 (JEE Main 2025)

Core Formula & Concept:

In electromagnetism, two fundamental dipole moments are defined:

• Electric dipole moment ($\vec{p}$): For two equal and opposite charges $+q$ and $-q$ separated by a displacement $\vec{d}$, the electric dipole moment is $\vec{p} = q \vec{d}.$ Its SI unit is coulomb-metre ($\text{C m}$).
• Magnetic dipole moment ($\vec{m}$): For a current loop of area $A$ carrying current $I$, the magnetic dipole moment is $\vec{m} = I \vec{A}.$ Its SI unit is ampere-square-metre ($\text{A m}^2$).

The question asks for the dimensions of the ratio $\frac{\text{Electric dipole moment}}{\text{Magnetic dipole moment}} = \frac{p}{m}.$ We must express both $p$ and $m$ in terms of the fundamental dimensions mass ($\text{M}$), length ($\text{L}$), time ($\text{T}$), and electric current ($\text{A}$), then compute the ratio.

Step-by-Step Derivation:

Step 1: Dimensions of electric dipole moment $p$

Charge $q$ has dimension $[\text{A T}]$, and displacement $d$ has dimension $[\text{L}]$. Hence $[p] = [q] \cdot [d] = \text{A T} \cdot \text{L} = \text{A L T}.$

Step 2: Dimensions of magnetic dipole moment $m$

Current $I$ has dimension $[\text{A}]$, and area $A$ has dimension $[\text{L}^2]$. Therefore $[m] = [I] \cdot [A] = \text{A} \cdot \text{L}^2 = \text{A L}^2.$

Step 3: Dimensions of the ratio $\frac{p}{m}$

Divide the two results: $\left[\frac{p}{m}\right] = \frac{[\text{A L T}]}{[\text{A L}^2]} = \frac{\text{A L T}}{\text{A L}^2} = \text{L}^{-1} \text{T}.$ In the standard form $[\text{M}^P \text{L}^Q \text{T}^R \text{A}^S]$, we read off $P = 0,\quad Q = -1,\quad R = 1,\quad S = 0.$ The question asks only for $P$ and $Q$, so the answer is $P=0$, $Q=-1$.

Common Traps & Exam Tip:

1. Confusing the definitions of $p$ and $m$: Students sometimes mix up the formulas for electric and magnetic dipole moments, leading to incorrect dimensions. 2. Sign errors in exponents: When dividing dimensions, it is easy to misplace a negative sign on the length exponent. 3. Overlooking the current dimension: One might forget that the magnetic dipole moment carries an extra factor of current, which cancels in the ratio but must be tracked carefully.

Exam Tip: Always write each quantity’s dimensions explicitly in terms of $\text{M},\text{L},\text{T},\text{A}$ before forming the ratio. This systematic approach avoids sign and exponent errors.