JEE PYQ: Units & Measurements - Question ID 4bb097bd6a85 (JEE Main 2008)

Core Formula & Concept:

To determine the dimensional formula of the magnetic field ($B$), we rely on the fundamental force law that connects electricity and magnetism: the Lorentz force. The Lorentz force on a charge $q$ moving with velocity $\vec{v}$ in a magnetic field $\vec{B}$ is given by: $\vec{F} = q \left( \vec{v} \times \vec{B} \right)$ From this, we can isolate the magnetic field $B$ and express it in terms of the dimensions of force ($F$), charge ($q$), and velocity ($v$).

Step-by-Step Derivation:

Step 1: Express the magnitude of the Lorentz force.
The magnitude of the Lorentz force is: $F = q v B \sin \theta$ Since $\sin \theta$ is dimensionless, we can ignore it for dimensional analysis. Thus: $F = q v B$

Step 2: Solve for the magnetic field $B$.
Rearranging the equation to solve for $B$: $B = \frac{F}{q v}$

Step 3: Substitute the dimensions of each quantity.
We know the dimensions of the quantities involved:

• Force ($F$): $MLT^{-2}$
• Charge ($q$): $C$ (Coulomb)
• Velocity ($v$): $LT^{-1}$

Substituting these into the equation for $B$: $B = \frac{MLT^{-2}}{C \cdot LT^{-1}}$

Step 4: Simplify the dimensions.
Simplify the expression by canceling out common terms: $B = \frac{MLT^{-2}}{C L T^{-1}} = \frac{M}{C} \cdot \frac{L}{L} \cdot \frac{T^{-2}}{T^{-1}} = M T^{-1} C^{-1}$ Thus, the dimensional formula of the magnetic field is: $[B] = MT^{-1}C^{-1}$

Step 5: Match with the given options.
Comparing the derived dimension $MT^{-1}C^{-1}$ with the options:

• A: $MLT^{-1}C^{-1}$ → Incorrect (extra $L$)
• B: $MT^{2}C^{-2}$ → Incorrect (wrong powers of $T$ and $C$)
• C: $MT^{-1}C^{-1}$ → Correct
• D: $MT^{-2}C^{-1}$ → Incorrect (wrong power of $T$)

Common Traps & Exam Tip:

Students often make the following mistakes in this question:

1. Confusing magnetic field with magnetic flux: The magnetic flux ($\Phi_B$) has dimensions $ML^2T^{-2}C^{-1}$, which is different from the magnetic field. Students sometimes mix these up.
2. Incorrectly applying the Lorentz force formula: Some forget that the force depends on the cross product of velocity and magnetic field, leading to incorrect dimensional cancellations.
3. Miscounting powers of $L$: In the simplification step, students may incorrectly cancel $L$ or retain it, leading to options like A (which includes an extra $L$).
4. Misremembering the unit of charge: Some students use $A$ (Ampere) instead of $C$ (Coulomb) for charge, which complicates the dimensional analysis unnecessarily.

Exam Tip: Always start from the fundamental force law (Lorentz force) when deriving dimensions for magnetic field. Double-check each step for dimensional consistency, especially the powers of $L$, $T$, and $C$.