Question

Question 1.3: Check that the ratio ke 2 /G memp is dimensionless. Look up a Table of Physical Constants and determine the value of this ratio. What does the ratio signify?

Class 12 - Physics - Electric Charges And Fields Electric Charges And Fields Page-46

Answer 1

The ratio of electrostatic force to the gravitational force between an electron and proton which is separated r distance from each other is given by
Fe/Fg = {Ke.e/r²}/{GMeMp/r²} = Ke²/GMeMp
∵Ratio of forces is dimensionless . Means, Fe/Fg is dimensionless.
∴ Ke²/GMeMp also will be dimensionless .
Hence, the ratio of Ke²/GMeMp is dimensionless .

Here, K = 9 × 10⁹ Nm²/C²
e = 1.6 × 10⁻¹⁹ C
G = 6.67 × 10⁻¹¹ Nm²/Kg²
Me = 9.1 × 10⁻³¹ Kg
Mp = 1.67 × 10⁻²⁷ Kg

Now, Ke²/GMeMp = {9 × 10⁹ × (1.6 × 10⁻¹⁹)²}/{6.67 × 10⁻¹¹ × 9.1 × 10⁻³¹ × 1.67 × 10⁻²⁷}
= 0.23 × 10⁴⁰ = 2.3 × 10³⁹

Answer 2

Explanation:

$\Large{\red{\underline{\underline{\sf{\blue{Solution:}}}}}}$

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The given ratio is $\sf \dfrac{ke^2}{Gm_ep_e}$

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Where,

$\sf \hookrightarrow\:G\:=\:Gravitational\:constant$

and it's unit is $\sf Nm^2kg^{-2}$

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$\sf \hookrightarrow\:m_e\:and\:m_p\:=$ Masses of Electron and proton

Their unit is kg

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$\sf \hookrightarrow$ e = electric charge

it's unit is C

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$\sf \hookrightarrow\: \epsilon_0\:=$ Permittivity of free space

it's unit is $\sf Nm^2C^{-2}$

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Therefore, unit of given ration,

$\sf \dfrac{ke^2}{Gm_ep_e}\:=\: \dfrac{[Nm^2C^{-2}]\,[C^{-2}]}{[Nm^2kg^{-2}]\,[kg]\,[kg]}$

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= $\sf M^0L^0T^0$

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Hence the given ratio is Dimensionless

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$\sf \hookrightarrow\:e\:=\:1.6\times 10^{-19}C$

$\sf \hookrightarrow\:G\:=\:6.67\times 10^{-19}C$

$\sf \hookrightarrow\:m_e\:=\:9.1\times 10^{-31}kg$

$\sf \hookrightarrow\:m_p\:=\:1.66\times 10^{-27}kg$

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Hence the numerical value of given ratio is

$\sf \dfrac{ke^2}{Gm_em_p}\:=\: \dfrac{9\times 10^9\times (1.6\times 10^{-19})^2}{6.67\times 10^{-11}\times 9.1\times 10^{-3}\times 1.6\times 10^{-22}}$

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$\sf \approx\,2.3\times 10^{39}$

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$\hookrightarrow$ This is the ratio of electric force to the gravitational force between a proton and an electron, keeping distance between them constant