P.A.M. Dirac, a great physicist of 20^ th century found that from the following basic constants, a number having dimensions of time can be constructed: 1. Charge on electron (e) 2. Permittivity of free space (e) 3. Mass of electron (m) 4. Mass of proton (m) 5. Speed of light (c) 6. Universal gravitational constant (G) Obtain Dirac's number, given that the desired number is proportional to m_p^ -1 and m^ -2 _e What is the significance of this number?

Let X be the desired number, Then: X=k e^ u _0^ x m^ -2 _e m^ -1 _ p c^ y G^ z (i) Here k is a dimensionless constant and x, y, z and u are unknowns, whose value is to be obtained from the principle of homogeneity of dimensions. Now [X]=[M^0L^0T^1Q^0] [e]=[M^0L^0T^0Q^1] [ _0]=[M^ -1 L^ -3 T^2Q] [C]=[M^0LT^ -1 ] [G]=[M^ -1 L^3T^ -2 ] Substituting dimensions of parameters involved in equation (i), We get: [M^0L^0T^1Q^0] =Q^ u [M^ -1 L^ -3 T^2Q^2]^ x M^ -2 M^ -1 [LT^ -1 ]^ y [M^ -1 L^3T^ -2 ]z =[M^ -x-3-z L^ -3x+y+3z T^ 2x-y+2z Q^ u+2x ] From the principle of homogeneity of dimensions -x - 3 - z = 0 ... (ii) - 3x + y + 3z = 0 (iii) 2x - y - 2z = 1 (iv) u + 2x = 0 (v) Solving equations. (ii), (iii), (iv) and (v) We get: u=+4,x=-2 y=-3 z=-1 x=ke^ 4 ^ -2 _0 m_e^ -2 m_ p ^ -1 c^ -3 G^ -1 Or x= ke^4 ^2_0 m^2_e m_ p c^3G Experiments show that, k=1 16 ^2 Hence, x= e^4 16 ^2 ^2_0 m^2_e m_pc^3G

P.A.M. Dirac, a great physicist of 20^ th century found that from…

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Question

$\text{P.A.M}$. Dirac, a great physicist of $20^{th}$ century found that from the following basic constants, a number having dimensions of time can be constructed:

Charge on electron $(e)$
Permittivity of free space $(e)$
Mass of electron $(m)$
Mass of proton $(m)$
Speed of light $(c)$
Universal gravitational constant $(G)$

Obtain Dirac's number, given that the desired number is proportional to $\text{m}_\text{p}^{-1}$ and $\text{m}^{-2}_\text{e}$ What is the significance of this number?

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Answer

Let $X$ be the desired number,
Then: $\text{X}=\text{k}\text{ e}^{\text{u }}\in_0^{\text{x}}\text{ m}^{-2}_\text{e}\text{ m}^{-1}_{\text{p}}\text{ c}^{\text{y}}\text{ G}^{\text{z}}\dots(\text{i})$
Here $k$ is a dimensionless constant and $x, y, z$ and $u$ are unknowns,
whose value is to be obtained from the principle of homogeneity of dimensions.
Now $[\text{X}]=[\text{M}^0\text{L}^0\text{T}^1\text{Q}^0]$
$[\text{e}]=[\text{M}^0\text{L}^0\text{T}^0\text{Q}^1]$
$[\in_0]=[\text{M}^{-1}\text{L}^{-3}\text{T}^2\text{Q}]$
$[\text{C}]=[\text{M}^0\text{L}\text{T}^{-1}]$
$[\text{G}]=[\text{M}^{-1}\text{L}^3\text{T}^{-2}]$
Substituting dimensions of parameters involved in equation $(i),$
We get: $[\text{M}^0\text{L}^0\text{T}^1\text{Q}^0]$
$=\text{Q}^{\text{u}}[\text{M}^{-1}\text{L}^{-3}\text{T}^2\text{Q}^2]^{\text{x}}\text{M}^{-2}\text{M}^{-1}[\text{LT}^{-1}]^{\text{y}}[\text{M}^{-1}\text{L}^3\text{T}^{-2}]\text{z}$
$=[\text{M}^{-\text{x}-3-\text{z}}\text{L}^{-3\text{x}+\text{y}+\text{3z}}\text{T}^{2\text{x}-\text{y}+\text{2z}}\text{Q}^{\text{u}+2\text{x}}]$
From the principle of homogeneity of dimensions $-\text{x} - 3 - \text{z} = 0 ... (\text{ii})$
$- 3\text{x} + \text{y} + 3\text{z} = 0 \dots (\text{iii})$
$2\text{x} - \text{y} - 2\text{z} = 1\dots(\text{iv})$
$\text{u} + 2\text{x} = 0\dots(\text{v})$ Solving equations. $(ii), (iii), (iv)$ and $(v)$
We get: $\text{u}=+4,\text{x}=-2\text{ y}=-3\text{ z}=-1$
$\therefore\text{x}=\text{ke}^{4}\in^{-2}_0\text{ m}_\text{e}^{-2}\text{ m}_{\text{p}}^{-1}\text{ c}^{-3}\text{G}^{-1}$ Or $\text{x}=\frac{\text{ke}^4}{\in^2_0\text{ m}^2_\text{e}\text{ m}_{\text{p}}\text{ c}^3\text{G}}$
Experiments show that, $\text{k}=\frac{1}{16\pi^2}$
Hence, $\text{x}=\frac{\text{e}^4}{16\pi^2\in^2_0\text{ m}^2_\text{e}\text{ m}_\text{p}\text{c}^3\text{G}}$