Question
If $$p$$ represents radiation pressure, $$c$$ represents speed of light and $$S$$ represents radiation energy striking unit area per $$\sec.$$ The non-zero integers $$x, y, z$$ such that $${p^x}{S^y}{c^z}$$ is dimensionless are
A.
$$x = 1,y = 1,z = 1$$
B.
$$x = - 1,y = 1,z = 1$$
C.
$$x = 1,y = - 1,z = 1$$
D.
$$x = 1,y = 1,z = - 1$$
Answer :
$$x = 1,y = - 1,z = 1$$
Solution :
Radiation pressure, $$p = \left[ {M{L^{ - 1}}{T^{ - 2}}} \right]$$
Velocity of light, $$c = \left[ {L{T^{ - 1}}} \right]$$
Energy striking unit area per second
$$S = \frac{{\left[ {M{L^2}{T^{ - 2}}} \right]}}{{\left[ {{L^2}T} \right]}} = \left[ {M{T^{ - 3}}} \right]$$
Now, $${p^x}{S^y}{c^z}$$ is dimensionless.
$$\eqalign{
& \therefore \left[ {{M^0}\;{L^0}\;{T^0}} \right] = {p^x}{S^y}{c^z} \cr
& {\text{or}}\,\left[ {{M^0}\;{L^0}\;{T^0}} \right] = {\left[ {{M^1}\;{L^{ - 1}}\;{T^{ - 2}}} \right]^x}{\left[ {{M^1}\;{T^{ - 3}}} \right]^y}{\left[ {\;{L^1}\;{T^{ - 1}}} \right]^z} \cr
& {\text{or}}\,\left[ {{M^0}\;{L^0}\;{T^0}} \right] = {\left[ M \right]^{x + y}}\;{\left[ L \right]^{ - x + z}}{\left[ T \right]^{ - 2x - 3y - z}} \cr} $$
From principle of homogeneity of dimensions
$$\eqalign{
& x + y = 0\,......\left( {\text{i}} \right) \cr
& - x + z = 0\,......\left( {{\text{ii}}} \right) \cr
& - 2x - 3y - z = 0\,......\left( {{\text{iii}}} \right) \cr} $$
Solving Eqs. (i), (ii) and (iii), we get
$$x = 1,y = - 1,z = 1$$