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Question Number 100068 by M±th+et+s last updated on 24/Jun/20
$${hello}\:{all}\:{i}\:{have}\:{some}\:{questions}? \\ $$$$ \\ $$$$\left.\mathrm{1}\right){what}\:{is}\:{the}\:{riemann}\:{hypothesis}? \\ $$$$ \\ $$$$\left.\mathrm{2}\right){how}\:{did}\:{they}\:{determine}\:{the}\:{distance} \\ $$$${to}\:{the}\:{sun}? \\ $$$$ \\ $$$$\left.\mathrm{3}\right){how}\:{did}\:{we}\:{measure}\:{the}\:{speed}\:{of}\:{light}? \\ $$$$ \\ $$
Answered by JDamian last updated on 24/Jun/20
$${You}\:{can}\:{find}\:{the}\:{answers}\:{in}\:{google} \\ $$
Commented by M±th+et+s last updated on 24/Jun/20
$${thank}\:{you}\:{for}\:{your}\:{comment}\:{sir}. \\ $$$${but} \\ $$$${in}\:{google}\:{there}\:{is}\:{no}\:{mathmatics}\: \\ $$$${explaning}\:{just}\:{a}\:{simple}\:{explanation}. \\ $$
Answered by smridha last updated on 25/Jun/20
![Maxwell diff:eq^n said us ▽^2 E^→ =𝛍_0 ε_0 (∂^2 E^→ /∂t^2 ) and ▽^2 B^→ =𝛍_0 𝛆_0 (∂^2 B^→ /∂t^2 ) where E^→ =electric field B^→ =magnetic field 𝛍_0 =magnetic permiability in free space 𝛆_0 =permitivity of free space [this is besically 3d wave eq^n this two eq^(ns) describe light as a electromagnetic wave] now analysis the dimention of the product of μ_0 and 𝛆_0 .. dim[𝛆_0 𝛍_0 ]=[M^(−1) L^(−1) T^4 ].[ML^(−1) T^(−2) ] =[L^(−2) T^2 ] now dim[(1/(√(𝛍_0 𝛆_0 )))]=[L.T^(−1) ]=dim(v) where v=velocity but this not any odinary velocity this is very special, the speed of light in free space denoted by C.. so our eq^n looks like: ▽^2 E^→ =(1/C^2 )(∂^2 E^→ /∂t^2 ) and ▽^2 B^→ =(1/C^2 ).(∂^2 B^→ /∂t^2 ) now speed of light: C=(1/(√(𝛍_0 𝛆_0 )))=(1/(√(4𝛑×10^(−7) ×8.854×10^(−12) ))) =2.99795638×10^8 ≈3×10^8 m.s^(−1) here I used the experimental value of 𝛍_0 and 𝛆_0 .](Q100083.png)
$$\:\boldsymbol{{M}}{a}\boldsymbol{{xwell}}\:\boldsymbol{{diff}}:\boldsymbol{{eq}}^{\boldsymbol{{n}}} \:\boldsymbol{{said}}\:\boldsymbol{{us}} \\ $$$$\bigtriangledown^{\mathrm{2}} \overset{\rightarrow} {\boldsymbol{{E}}}=\boldsymbol{\mu}_{\mathrm{0}} \epsilon_{\mathrm{0}} \frac{\partial^{\mathrm{2}} \overset{\rightarrow} {\boldsymbol{{E}}}}{\partial{t}^{\mathrm{2}} }\:\boldsymbol{{and}}\:\bigtriangledown^{\mathrm{2}} \overset{\rightarrow} {\boldsymbol{{B}}}=\boldsymbol{\mu}_{\mathrm{0}} \boldsymbol{\epsilon}_{\mathrm{0}} \frac{\partial^{\mathrm{2}} \overset{\rightarrow} {\boldsymbol{{B}}}}{\partial{t}^{\mathrm{2}} } \\ $$$$\boldsymbol{{where}}\:\overset{\rightarrow} {\boldsymbol{{E}}}=\boldsymbol{{electric}}\:\boldsymbol{{field}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\overset{\rightarrow} {\boldsymbol{{B}}}=\boldsymbol{{magnetic}}\:\boldsymbol{{field}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\mu}_{\mathrm{0}} =\boldsymbol{{magnetic}}\:\boldsymbol{{permiability}}\:\boldsymbol{{in}}\:\boldsymbol{{free}}\:\boldsymbol{{space}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\epsilon}_{\mathrm{0}} =\boldsymbol{{permitivity}}\:\boldsymbol{{of}}\:\boldsymbol{{free}}\:\boldsymbol{{space}} \\ $$$$\left[\boldsymbol{{this}}\:\boldsymbol{{is}}\:\boldsymbol{{besically}}\:\mathrm{3}\boldsymbol{{d}}\:\boldsymbol{{wave}}\:\boldsymbol{{eq}}^{\boldsymbol{{n}}} \right. \\ $$$$\boldsymbol{{this}}\:\boldsymbol{{two}}\:\boldsymbol{{eq}}^{\boldsymbol{{ns}}} \:\boldsymbol{{describe}}\:\boldsymbol{{light}}\:\boldsymbol{{as}} \\ $$$$\left.\boldsymbol{{a}}\:\boldsymbol{{electromagnetic}}\:\boldsymbol{{wave}}\right] \\ $$$$\:\:\boldsymbol{{now}}\:\boldsymbol{{analysis}}\:\boldsymbol{{the}}\:\boldsymbol{{dimention}}\:\boldsymbol{{of}} \\ $$$$\boldsymbol{{the}}\:\boldsymbol{{product}}\:\boldsymbol{{of}}\:\mu_{\mathrm{0}} \:\boldsymbol{{and}}\:\boldsymbol{\epsilon}_{\mathrm{0}} .. \\ $$$$\:\:\:\boldsymbol{{dim}}\left[\boldsymbol{\epsilon}_{\mathrm{0}} \boldsymbol{\mu}_{\mathrm{0}} \right]=\left[\boldsymbol{{M}}^{−\mathrm{1}} \boldsymbol{{L}}^{−\mathrm{1}} \boldsymbol{{T}}^{\mathrm{4}} \right].\left[\boldsymbol{{ML}}^{−\mathrm{1}} \boldsymbol{{T}}^{−\mathrm{2}} \right] \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\left[\boldsymbol{{L}}^{−\mathrm{2}} \boldsymbol{{T}}^{\mathrm{2}} \right] \\ $$$$\boldsymbol{{now}}\:\boldsymbol{{dim}}\left[\frac{\mathrm{1}}{\sqrt{\boldsymbol{\mu}_{\mathrm{0}} \boldsymbol{\epsilon}_{\mathrm{0}} }}\right]=\left[\boldsymbol{{L}}.\boldsymbol{{T}}^{−\mathrm{1}} \right]=\boldsymbol{{dim}}\left(\boldsymbol{{v}}\right) \\ $$$$\boldsymbol{{where}}\:\boldsymbol{{v}}=\boldsymbol{{velocity}}\:\boldsymbol{{but}}\:\boldsymbol{{this}}\:\boldsymbol{{not}} \\ $$$$\boldsymbol{{any}}\:\boldsymbol{{odinary}}\:\boldsymbol{{velocity}}\:\boldsymbol{{this}}\:\boldsymbol{{is}}\:\boldsymbol{{very}} \\ $$$$\boldsymbol{{special}},\:\boldsymbol{{the}}\:\boldsymbol{{speed}}\:\boldsymbol{{of}}\:\boldsymbol{{light}}\:\boldsymbol{{in}} \\ $$$$\boldsymbol{{free}}\:\boldsymbol{{space}}\:\boldsymbol{{denoted}}\:\boldsymbol{{by}}\:\boldsymbol{{C}}.. \\ $$$$\boldsymbol{{so}}\:\boldsymbol{{our}}\:\boldsymbol{{eq}}^{\boldsymbol{{n}}} \:\boldsymbol{{looks}}\:\boldsymbol{{like}}: \\ $$$$\bigtriangledown^{\mathrm{2}} \overset{\rightarrow} {\boldsymbol{{E}}}=\frac{\mathrm{1}}{\boldsymbol{{C}}^{\mathrm{2}} }\frac{\partial^{\mathrm{2}} \overset{\rightarrow} {\boldsymbol{{E}}}}{\partial\boldsymbol{{t}}^{\mathrm{2}} }\:{and}\:\bigtriangledown^{\mathrm{2}} \overset{\rightarrow} {\boldsymbol{{B}}}=\frac{\mathrm{1}}{\boldsymbol{{C}}^{\mathrm{2}} }.\frac{\partial^{\mathrm{2}} \overset{\rightarrow} {\boldsymbol{{B}}}}{\partial\boldsymbol{{t}}^{\mathrm{2}} } \\ $$$$\boldsymbol{{now}}\:\boldsymbol{{speed}}\:\boldsymbol{{of}}\:\boldsymbol{{light}}: \\ $$$$\:\:\boldsymbol{{C}}=\frac{\mathrm{1}}{\sqrt{\boldsymbol{\mu}_{\mathrm{0}} \boldsymbol{\epsilon}_{\mathrm{0}} }}=\frac{\mathrm{1}}{\sqrt{\mathrm{4}\boldsymbol{\pi}×\mathrm{10}^{−\mathrm{7}} ×\mathrm{8}.\mathrm{854}×\mathrm{10}^{−\mathrm{12}} }} \\ $$$$\:\:\:\:\:=\mathrm{2}.\mathrm{99795638}×\mathrm{10}^{\mathrm{8}} \approx\mathrm{3}×\mathrm{10}^{\mathrm{8}} \boldsymbol{{m}}.\boldsymbol{{s}}^{−\mathrm{1}} \\ $$$$\boldsymbol{{here}}\:\boldsymbol{{I}}\:\boldsymbol{{used}}\:\boldsymbol{{the}}\:\boldsymbol{{experimental}}\:\boldsymbol{{value}} \\ $$$$\boldsymbol{{of}}\:\boldsymbol{\mu}_{\mathrm{0}} \:\boldsymbol{{and}}\:\boldsymbol{\epsilon}_{\mathrm{0}} . \\ $$
Commented by Rasheed.Sindhi last updated on 25/Jun/20
$$\mathcal{G}\overset{\frown} {\mathcal{O}}\overset{\frown} {\mathcal{O}D}\:{use}\:{of}\:{mathematical}\:{notation} \\ $$$${in}\:{the}\:\boldsymbol{{field}}\:{of}\:\boldsymbol{{human}}\:\boldsymbol{{emotions}}! \\ $$
Commented by M±th+et+s last updated on 24/Jun/20
$${very}\:{nice}\:{explanation}\:. \\ $$$${god}\:{bless}\:{you} \\ $$
Commented by smridha last updated on 24/Jun/20
$$\underset{\boldsymbol{{i}}=\mathrm{1}} {\overset{\infty} {\sum}}\left(\boldsymbol{{thank}}\right)_{\boldsymbol{{i}}} \\ $$
Commented by Rasheed.Sindhi last updated on 25/Jun/20
$${Can}\:{we}\:{consider}\: \\ $$$$\boldsymbol{{human}}-\boldsymbol{{emotions}}-\boldsymbol{{field}} \\ $$$${as}\:{a}\:\boldsymbol{{mathematical}}-\boldsymbol{{field}}? \\ $$$${Of}\:{course},\:{why}\:{not} \\ $$$$\boldsymbol{{if}}\:\:{we}\:{can}\:\boldsymbol{{define}}\:\boldsymbol{{thanks}}, \\ $$$$\boldsymbol{{sorry}}\:\&\:{other}\:{human}-{emotions}! \\ $$
Commented by smridha last updated on 25/Jun/20
$$\boldsymbol{{yeah}}\:\boldsymbol{{why}}\:\boldsymbol{{not}}!!\boldsymbol{{after}}\:\boldsymbol{{all}}\: \\ $$$$\boldsymbol{{M}}{a}\boldsymbol{{thematics}}\:\boldsymbol{{is}}\:\boldsymbol{{nothing}}\:\boldsymbol{{more}}\:\boldsymbol{{than}}\: \\ $$$$\boldsymbol{{a}}\:\boldsymbol{{language}}.\boldsymbol{{if}}\:\boldsymbol{{there}}\:\boldsymbol{{is}}\:\boldsymbol{{a}}\:\boldsymbol{{wish}} \\ $$$$\boldsymbol{{there}}\:\boldsymbol{{is}}\:\boldsymbol{{a}}\:\boldsymbol{{way}}.. \\ $$