question_answer

If p represents radian pressure, c represents the speed of light and q represents the radiation energy, per unit area per unit time, then calculate non-zero integers such that \[{{p}^{x}}{{q}^{y}}{{c}^{z}}\] is dimensionless.

Answer:

                Here,     \[\text{p }=\frac{ML{{T}^{-2}}}{{{L}^{2}}}=[M{{L}^{-1}}{{T}^{-2}}]\] \[\text{c}=\left[ \text{L}{{\text{T}}^{-\text{1}}} \right]\] \[\text{q }=\frac{energy}{area\times time}=\frac{M{{L}^{2}}{{T}^{-2}}}{{{L}^{2}}T}=[M{{T}^{-3}}]\] \[\therefore \]   \[{{\text{p}}^{\text{x}}}{{\text{q}}^{\text{y}}}{{\text{c}}^{\text{z}}}=\text{ }{{\left( \text{M}{{\text{L}}^{-\text{1}}}{{\text{T}}^{-\text{2}}} \right)}^{\text{x}}}{{\left( \text{M}{{\text{T}}^{-\text{3}}} \right)}^{\text{y}}}{{\left( \text{L}{{\text{T}}^{-\text{1}}} \right)}^{\text{z}}}=\text{ }{{\text{M}}^{\text{x }+\text{ y}}}{{\text{L}}^{-\text{x }+\text{ z}}}{{\text{T}}^{-\text{2x}}}^{-\text{3y}-\text{z}}\] It will be dimensionless, if            \[\text{x}+\text{y}=0\] \[\therefore \] \[x=-y\] \[\text{x }+\text{ z }=0\]and \[\text{x}=\text{z}\text{2x}\text{3y}-\text{z}=0\] These equations have infinite solutions. One of the solutions is x = 1,y = -1 and z = 1, so that \[{{p}^{x}}{{q}^{y}}{{c}^{z}}\]isdimensionless.