QConsider the equation $H=\dfrac{x^p\cdot\varepsilon^q\cdot E^r}{t^s}$ where $H$ = magnetic field intensity, $E$ = electric field, $\varepsilon$ = permittivity, $x$ = distance, $t$ = time. The values of $p,q,r,s$ respectively are:
A) 1,1,1,1 B) −1,1,2,1 C) 1,−1,−2,1 D) −1,−2,−2,1Units & Dimensions
MCQ
1
Write dimensions of each quantity
$[H]=\text{A m}^{-1}=[\text{M}^0\text{L}^{-1}\text{T}^0\text{A}^1]$
$[E]=\text{V m}^{-1}=[\text{M}^1\text{L}^1\text{T}^{-3}\text{A}^{-1}]$
$[\varepsilon]=\text{C}^2\text{N}^{-1}\text{m}^{-2}=[\text{M}^{-1}\text{L}^{-3}\text{T}^4\text{A}^2]$
$[x]=[\text{L}]$, $[t]=[\text{T}]$
2
Set up dimensional equations
$[\text{M}^0\text{L}^{-1}\text{T}^0\text{A}^1]=[\text{L}]^p\cdot[\text{M}^{-1}\text{L}^{-3}\text{T}^4\text{A}^2]^q\cdot[\text{M}\text{L}\text{T}^{-3}\text{A}^{-1}]^r\cdot[\text{T}]^{-s}$
Matching each dimension:
M: $0=-q+r\Rightarrow q=r$
L: $-1=p-3q+r$
T: $0=4q-3r-s$
A: $1=2q-r$
From $q=r$ and $2q-r=1$: $2q-q=1\Rightarrow q=1,r=1$
L: $-1=p-3+1=p-2\Rightarrow p=1$
T: $0=4-3-s=1-s\Rightarrow s=1$
3
Result
$$p=1,\quad q=1,\quad r=1,\quad s=1\quad\Rightarrow\boxed{(A)}$$
📘 Dimensional Analysis Method
Write dimensions of all quantities in M, L, T, A (mass, length, time, current). Equate powers on both sides for each base dimension separately to get a system of equations. Here 4 equations in 4 unknowns yield unique solution.
Dimensions of H FieldH (magnetic field intensity) = A/m = [M⁰L⁻¹T⁰A¹]. Note: B (magnetic flux density) = [ML⁻²T⁻²A⁻¹] is different.
Dimensions of ε (permittivity)ε₀ = C²/(N·m²) = C²·s²/(kg·m³) = [M⁻¹L⁻³T⁴A²]. From F=q²/(4πε₀r²): [ε]=[C²/(N·m²)]=[A²T²/(kg·m/s²·m²)].
Dimensions of E (electric field)E = V/m = N/C = [kg·m/(A·s³)] = [MLT⁻³A⁻¹].
Cross-Check with MaxwellThe equation ∇×H=εoE/∂t relates H, ε, E, x, t. Since [∇×H]=[H/x] and [ε∂E/∂t]=[ε·E/t]: [H/x]=[εE/t], giving H=εEx/t — consistent with p=1,q=1,r=1,s=1.
1
What are the dimensions of magnetic field H?
⌄
H field: [A/m] = [M⁰L⁻¹T⁰A¹]. Don't confuse with B field which has units Tesla.
2
What are the dimensions of permittivity ε?
⌄
[ε] = [C²/(N·m²)] = [M⁻¹L⁻³T⁴A²].
3
Is there a physical equation that matches H=xεE/t?
⌄
Yes! From Maxwell's equations: ∂H/∂x = ε₀∂E/∂t in free space. Rearranging: H = ε₀·E·x/t (roughly), consistent with p=q=r=s=1.
4
Why check M equation first?
⌄
The M equation (−q+r=0) immediately gives q=r, which simplifies all other equations.
5
Is this from JEE Main 2026?
⌄
Yes, this appeared in JEE Main 2026.