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

Now, substitute these dimensions into the ratio $\frac{ke^2}{Gm_e m_p}$:

$$\left[\frac{ke^2}{Gm_e m_p}\right] = \frac{[k][e^2]}{[G][m_e m_p]}$$ $$= \frac{[\text{ML}^3 \text{T}^{-4} \text{A}^{-2}][\text{A}^2 \text{T}^2]}{[\text{M}^{-1} \text{L}^3 \text{T}^{-2}][\text{M}^2]}$$

Combine the terms in the numerator:

$$[\text{ML}^3 \text{T}^{-4} \text{A}^{-2}][\text{A}^2 \text{T}^2] = [\text{M}^1 \text{L}^3 \text{T}^{-4+2} \text{A}^{-2+2}] = [\text{ML}^3 \text{T}^{-2} \text{A}^0] = [\text{ML}^3 \text{T}^{-2}]$$

Combine the terms in the denominator:

$$[\text{M}^{-1} \text{L}^3 \text{T}^{-2}][\text{M}^2] = [\text{M}^{-1+2} \text{L}^3 \text{T}^{-2}] = [\text{M}^1 \text{L}^3 \text{T}^{-2}]$$

Substitute these back into the ratio:

$$\left[\frac{ke^2}{Gm_e m_p}\right] = \frac{[\text{ML}^3 \text{T}^{-2}]}{[\text{M}^1 \text{L}^3 \text{T}^{-2}]}$$

Simplify the exponents:

$$= [\text{M}^{1-1} \text{L}^{3-3} \text{T}^{-2-(-2)}] = [\text{M}^0 \text{L}^0 \text{T}^0]$$

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 $k = 8.98755 \times 10^9 \, \text{N m}^2/\text{C}^2$
• Elementary charge $e = 1.602176634 \times 10^{-19} \, \text{C}$
• Gravitational constant $G = 6.67430 \times 10^{-11} \, \text{N m}^2/\text{kg}^2$
• 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 $ke^2$:

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

Calculate the denominator $Gm_e m_p$:

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