JEE PYQ: Units & Measurements - Question ID c432df107932 (JEE Main 2020)

In electromagnetism, the expression ${B^2 \over 2\mu_0}$ represents the energy density stored in a magnetic field $B$ in vacuum. The constant $\mu_0$ is the permeability of free space.

To find its dimensions, we recall:

• Force between two current-carrying wires gives the dimension of $\mu_0$: $F = {\mu_0 I_1 I_2 \ell \over 2\pi r} \quad\Longrightarrow\quad [\mu_0] = { [F]\,[r] \over [I]^2\,[\ell] }.$
• Magnetic field $B$ is defined via the Lorentz force $F = q\,v\,B$, so $[B] = { [F] \over [q]\,[v] }.$

Step 1: Dimensions of $\mu_0$

From the force formula $F = {\mu_0 I^2 \ell \over 2\pi r}$ we isolate $\mu_0$: $\mu_0 = {2\pi r\,F \over I^2\,\ell}.$ Taking dimensions: $[\mu_0] = { [r]\,[F] \over [I]^2\,[\ell] } = { \mathrm{L}\;\mathrm{MLT}^{-2} \over \mathrm{A}^2\;\mathrm{L} } = \mathrm{MLT}^{-2}\,\mathrm{A}^{-2}.$

Step 2: Dimensions of $B$

From $F = q\,v\,B$ we get $B = {F \over q\,v}.$ Hence $[B] = { [F] \over [q]\,[v] } = { \mathrm{MLT}^{-2} \over \mathrm{AT}\;\mathrm{LT}^{-1} } = \mathrm{MT}^{-2}\,\mathrm{A}^{-1}.$

Step 3: Dimensions of $B^2$

Square the dimension of $B$: $[B^2] = \bigl(\mathrm{MT}^{-2}\,\mathrm{A}^{-1}\bigr)^2 = \mathrm{M}^2\,\mathrm{T}^{-4}\,\mathrm{A}^{-2}.$

Step 4: Combine to find $\displaystyle{B^2 \over 2\mu_0}$

Divide $[B^2]$ by $[\mu_0]$: $$\left[{B^2 \over \mu_0}\right] = { \mathrm{M}^2\,\mathrm{T}^{-4}\,\mathrm{A}^{-2} \over \mathrm{MLT}^{-2}\,\mathrm{A}^{-2} } = \mathrm{M}^{2-1}\,\mathrm{L}^{-1}\,\mathrm{T}^{-4+2} = \mathrm{ML}^{-1}\,\mathrm{T}^{-2}.$$ The factor of $2$ is dimensionless, so the final dimension is $\boxed{\mathrm{ML}^{-1}\,\mathrm{T}^{-2}}.$

1. Confusing $\mu_0$ with $\epsilon_0$: Students sometimes use the dimensions of permittivity $\epsilon_0$ instead of permeability $\mu_0$. 2. Forgetting the square on $B$: Omitting the exponent $2$ leads to an incorrect power of mass or time. 3. Sign errors in exponents: Careful bookkeeping of negative signs in time and length exponents is essential.

Tip: Always write each step dimensionally and cancel units systematically to avoid arithmetic slips.