Check that the ratio ke2 /Gme mp is dimensionless. Look up a table of physical constants and determine the value of this ratio. What does the ratio signify?

Step 1: Check that the ratio $\frac{ke^2}{Gm_e m_p}$ is dimensionless.

First, determine the dimensions of each constant and quantity: • Dimensions of Coulomb's constant $k$: From Coulomb's Law, $F = k \frac{q_1 q_2}{r^2}$, so $k = \frac{F r^2}{q_1 q_2}$. $[k] = \frac{[MLT^{-2}][L^2]}{[AT][AT]} = \frac{[ML^3 T^{-2}]}{[A^2 T^2]} = [ML^3 T^{-4} A^{-2}]$ • Dimensions of elementary charge $e$: The dimension of charge is $[AT]$. $[e] = [AT]$ So, the dimension of $e^2$ is: $[e^2] = [A^2 T^2]$ • Dimensions of gravitational constant $G$: From Newton's Law of Gravitation, $F = G \frac{m_1 m_2}{r^2}$, so $G = \frac{F r^2}{m_1 m_2}$. $[G] = \frac{[MLT^{-2}][L^2]}{[M][M]} = \frac{[ML^3 T^{-2}]}{[M^2]} = [M^{-1} L^3 T^{-2}]$ • Dimensions of electron mass $m_e$ and proton mass $m_p$: The dimension of mass is $[M]$. $[m_e] = [M]$ $[m_p] = [M]$ So, the dimension of $m_e m_p$ is: $[m_e m_p] = [M^2]$

Now, substitute these dimensions into the ratio $\frac{ke^2}{Gm_e m_p}$: $\left[\frac{ke^2}{Gm_e m_p}\right] = \frac{[k][e^2]}{[G][m_e m_p]}$ $= \frac{[ML^3 T^{-4} A^{-2}][A^2 T^2]}{[M^{-1} L^3 T^{-2}][M^2]}$ Combine the terms in the numerator: $[ML^3 T^{-4} A^{-2}][A^2 T^2] = [M^1 L^3 T^{-4+2} A^{-2+2}] = [ML^3 T^{-2} A^0] = [ML^3 T^{-2}]$ Combine the terms in the denominator: $[M^{-1} L^3 T^{-2}][M^2] = [M^{-1+2} L^3 T^{-2}] = [M^1 L^3 T^{-2}]$ Substitute these back into the ratio: $\left[\frac{ke^2}{Gm_e m_p}\right] = \frac{[ML^3 T^{-2}]}{[M^1 L^3 T^{-2}]}$ Simplify the exponents: $= [M^{1-1} L^{3-3} T^{-2-(-2)}] = [M^0 L^0 T^0]$ Since all exponents are zero, the ratio is dimensionless.

Step 2: Determine the value of this ratio. Use the following physical constants: • Coulomb's constant $k = 8.98755 \times 10^9 \, N m^2/C^2$ • Elementary charge $e = 1.602176634 \times 10^{-19} \, \text{C}$ • Gravitational constant $G = 6.67430 \times 10^{-11} \, N m^2/kg^2$ • Electron mass $m_e = 9.1093837015 \times 10^{-31} \, \text{kg}$ • Proton mass $m_p = 1.67262192369 \times 10^{-27} \, \text{kg}$

Calculate the numerator $ke^2$: $ke^2 = (8.98755 \times 10^9 \, N m^2/C^2) \times (1.602176634 \times 10^{-19} \, C)^2$ $ke^2 = (8.98755 \times 10^9) \times (2.5669699 \times 10^{-38}) \, N m^2$ $ke^2 = 2.307077 \times 10^{-28} \, N m^2$

Calculate the denominator $Gm_e m_p$: