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Step-by-Step Solution
Step 1: Identify the Forces Involved
Two hydrogen atoms are placed at a large distance d apart. Each hydrogen atom consists of a proton and an electron. However, if the electron and proton charges differ by a small amount, one has charge −e and the other (e + Δe). The forces between the atoms will be:
The electrostatic force due to the net charge difference.
The gravitational force due to their masses.
Step 2: Express the Electrostatic Force
The electrostatic force FE between two charges Δe (since the net effective charge difference is Δe) separated by distance d is given by Coulomb’s Law:
$ F_{E} = \frac{1}{4 \pi \varepsilon_0} \frac{(\Delta e)^2}{d^2} $
Step 3: Express the Gravitational Force
The gravitational force FG between two masses mh (hydrogen atoms each of mass 1.67 × 10−27 kg) separated by the same distance d is given by Newton’s law of gravitation:
$ F_{G} = G \frac{m_{h}^2}{d^2} $
Step 4: Equate the Forces
According to the problem statement, these forces balance each other, so:
$ F_{E} = F_{G} $
Substitute the expressions from steps 2 and 3:
$ \frac{1}{4 \pi \varepsilon_0} \frac{(\Delta e)^2}{d^2} = G \frac{m_{h}^2}{d^2} $
We observe that (1 / d2) cancels out from both sides, leaving:
$ \frac{1}{4 \pi \varepsilon_0} (\Delta e)^2 = G m_{h}^2 $
Therefore:
$ \Delta e = \sqrt{ 4 \pi \varepsilon_0 \cdot G m_{h}^2 } $
Step 5: Simplify and Approximate
Using the approximate values:
$ 4 \pi \varepsilon_0 = \frac{1}{k} $ where $ k \approx 9 \times 10^9 \,\mathrm{N\,m^2\,C^{-2}}. $
$ G \approx 6.67 \times 10^{-11} \,\mathrm{N\,m^2\,kg^{-2}}, \quad m_{h} \approx 1.67 \times 10^{-27} \,\mathrm{kg}. $
Thus,
$ \Delta e = m_{h} \sqrt{\frac{G}{k}}
= 1.67 \times 10^{-27} \sqrt{\frac{6.67 \times 10^{-11}}{9 \times 10^{9}}} $
Carrying out the numerical computation gives a value on the order of:
$ \Delta e \approx 10^{-37}\,\mathrm{C}. $
Hence, Δe is of the order of 10−37 C.
