According to the theory of relativity, mass can be transformed into energy. The resulting energy E is proportional to certain powers of the mass m and the speed of light c . Determine a relation among these quantities by applying the method of dimensions.

Step-by-step solution

Let E = k m^a c^b , where k is a dimensionless constant. Writing the dimensional formula of each physical quantity: [E] = [M L^2 T⁻²] [m] = [M] [c] = [L T⁻¹] Substituting the dimensions in the assumed relation: [M L^2 T⁻²] = [M]^a [L T⁻¹]^b [M L^2 T⁻²] = [M^a L^b T^ -b ] Equating the powers of M , L , and T on both sides: a = 1 b = 2 -b = -2 b = 2 Substituting the values of a and b in the assumed relation: E = k m^1 c^2 E = k m c^2