JEE Main 2025 Jan 24 Shift 2, Physics Q17: Wave Nature of Matter

JEE Main2025Jan 24, Shift 2Physics

The energy $E$ and momentum $p$ of a moving body of mass $m$ are related by some equation. Given that $c$ represents the speed of light, identify the correct equation.

A
$E^{2} = p c^{2} + m^{2} c^{4}$
B
$E^{2} = p c^{2} + m^{2} c^{2}$
C
$E^{2} = p^{2} c^{2} + m^{2} c^{2}$
D
$E^{2} = p^{2} c^{2} + m^{2} c^{4}$

Solution

This is the well-known relativistic energy-momentum relation:

$E^{2} = p^{2} c^{2} + m^{2} c^{4}$

A quick dimensional check confirms it. With $[E] = M L^{2} T^{- 2}$ , $[p] = M L T^{- 1}$ , $[c] = L T^{- 1}$ :

$[p^{2} c^{2}] = (M L T^{- 1})^{2} (L T^{- 1})^{2} = M^{2} L^{4} T^{- 4} = [E^{2}]$ ✓
$[m^{2} c^{4}] = M^{2} (L T^{- 1})^{4} = M^{2} L^{4} T^{- 4} = [E^{2}]$ ✓

Both terms on the right have the dimensions of $E^{2}$ , while options (1)-(3) have dimensional mismatches.

Correct option: (4)

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JEE Main 2025 Jan 24 Shift 2, Physics Q17: Wave Nature of Matter

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