Question

Theory of relativity reveals that mass can be converted into energy. The energy E so obtained is proportional to certain powers of mass m and the speed c of light. Guess a relation among the quantities using the method of dimensions.

Concept of Physics - 1 , HC VERMA , Chapter "Introduction to Physics".

Answer 1

Hello Dear.


As per as the Question,

Energy ∝ (Mass)ˣ × (Speed of the light)ᵃ
where, ˣ and ᵃ is the powers.

∴ Energy = Contant × (m)ˣ  × (c)ᵃ
Let us taking the Constant as Y.

Now,
Finding the Dimensions of the Energy,

Energy = Force × Displacement.
 = Mass × Acceleration × Displacement.
 = Mass × Velocity/Time × Displacement.
 = M L T⁻² × L.
 = M L² T⁻².


Now,
Dimension of Mass = M.
Dimension of Speed of the Light = L T⁻¹.

∴ M L² T⁻² =  (M)ˣ  × (LT⁻¹)ᵃ
Equating the Exponents of M , L and T on both the Sides,

We get,

x = 1, and a = 2.

∴ Relation will be,
Energy = Y × m¹ × c²
Energy = Y mc².



Hope it helps.