Check that the ratio ke2/G memp is dimensionless. Look up a Tableof Physical Constants and determine the value of this ratio. Whatdoes the ratio signify?
Question
Check that the ratio ke2/G memp is dimensionless.
Look up a Table of Physical Constants and determine the value of this ratio.
What does the ratio signify?
Solution
First, let's check if the ratio ke2/Gmemp is dimensionless.
- ke is Coulomb's constant, which has units of N.m^2/C^2 (Newton meter squared per Coulomb squared).
- e is the elementary charge, which has units of C (Coulombs).
- G is the gravitational constant, which has units of m^3/kg.s^2 (meter cubed per kilogram per second squared).
- me and mp are the masses of the electron and proton respectively, which have units of kg (kilograms).
So, the units of ke2 are (N.m^2/C^2) * (C)^2 = N.m^2. The units of Gmemp are (m^3/kg.s^2) * (kg)^2 = m^3/s^2.
The ratio ke2/Gmemp therefore has units of (N.m^2) / (m^3/s^2) = s^2/N, which is not dimensionless.
Now, let's determine the value of this ratio.
- The value of ke is approximately 8.99 x 10^9 N.m^2/C^2.
- The value of e is approximately 1.60 x 10^-19 C.
- The value of G is approximately 6.67 x 10^-11 m^3/kg.s^2.
- The value of me is approximately 9.11 x 10^-31 kg.
- The value of mp is approximately 1.67 x 10^-27 kg.
So, the value of ke2 is (8.99 x 10^9 N.m^2/C^2) * (1.60 x 10^-19 C)^2 = 2.30 x 10^-28 N.m^2. The value of Gmemp is (6.67 x 10^-11 m^3/kg.s^2) * (9.11 x 10^-31 kg) * (1.67 x 10^-27 kg) = 1.02 x 10^-68 m^3/s^2.
The ratio ke2/Gmemp is therefore (2.30 x 10^-28 N.m^2) / (1.02 x 10^-68 m^3/s^2) = 2.25 x 10^40 s^2/N.
The ratio signifies the relative strength of the electromagnetic force to the gravitational force between an electron and a proton. A larger ratio means that the electromagnetic force is much stronger than the gravitational force.
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