Unità di misura di Planck derivate

Le unità di misura di Planck derivate sono quelle unità di misura derivate dalla combinazione delle unità di Planck fondamentali, come la lunghezza, la massa e il tempo.

Tabella[modifica | modifica wikitesto]

Unità derivate di Planck approssimate
Dimensione	Formula	Espressione		Valore, nel SI approssimata
Dimensione	Formula	Versione di Lorentz–Heaviside^[1]	Versione gaussiana^[2]^[3]^[4]^[5]	Valore nel SI Lorentz-Heaviside	Valore nel SI Gaussiana
Proprietà meccanico-fisiche
Area di Planck	Area $\left[L\right]^{2}$	$l_{\text{P}}^{2}={\frac {4\pi \hbar G}{c^{3}}}$	$l_{\text{P}}^{2}={\frac {\hbar G}{c^{3}}}$	$3,282688\cdot 10^{-69}\;m^{2}$	$2,612280\cdot 10^{-70}\;m^{2}$
Volume di Planck	Volume $\left[L\right]^{3}$	$l_{\text{P}}^{3}={\sqrt {\frac {64\pi ^{3}\hbar ^{3}G^{3}}{c^{9}}}}$	$l_{\text{P}}^{3}=\left({\frac {\hbar G}{c^{3}}}\right)^{\frac {3}{2}}={\sqrt {\frac {\hbar ^{3}G^{3}}{c^{9}}}}$	$1,880808\cdot 10^{-103}\;m^{3}$	$4,222111\cdot 10^{-105}\;m^{3}$
Velocità di Planck	Velocità $\left[L\right]\left[T\right]^{-1}$	$v_{\text{P}}={\frac {l_{\text{P}}}{t_{\text{P}}}}=c$		$299.792.458\;{\frac {m}{s}}$
Planck Angolare	Radiante $\left[L\right]\left[L\right]^{-1}\to$ adimensionale	$\theta _{\text{P}}={\frac {l_{\text{P}}}{l_{\text{P}}}}=1$		$1\;\mathrm {rad}$
Planck steradiante	Angolo solido $\left[L\right]^{2}\left[L\right]^{-2}\to$ adimensionale	$\theta _{\text{P}}^{2}={\frac {l_{\text{P}}^{2}}{l_{\text{P}}^{2}}}=1$		$1\;\mathrm {sr}$
Quantità di moto di Planck	Quantità di moto $\left[L\right]\left[M\right]\left[T\right]^{-1}$	$m_{\text{P}}c={\frac {\hbar }{l_{\text{P}}}}={\sqrt {\frac {\hbar c^{3}}{4\pi G}}}$	$m_{\text{P}}c={\frac {\hbar }{l_{\text{P}}}}={\sqrt {\frac {\hbar c^{3}}{G}}}$	$1,840608\;\mathrm {N} \cdot s$	$6,524785\;kg\cdot {\frac {m}{s}}$
Energia di Planck	Energia $\left[M\right]\left[L\right]^{2}\left[T\right]^{-2}$	$E_{\text{P}}=m_{\text{P}}v_{\text{P}}^{2}={\frac {\hbar }{t_{\text{P}}}}={\sqrt {\frac {\hbar c^{5}}{4\pi G}}}$	$E_{\text{P}}={{m}_{\text{P}}}{{c}^{2}}={\frac {\hbar }{{t}_{\text{P}}}}={\sqrt {\frac {\hbar {{c}^{5}}}{G}}}$	$5.518004\cdot 10^{8}\;\mathrm {J}$ $153,278\;k\mathrm {W} \cdot h$ $3,444067\cdot 10^{18}Ge\mathrm {V}$	$1,956081\cdot 10^{9}\mathrm {J}$ $543,356\;k\mathrm {W} \cdot h$ $1,220890\cdot 10^{28}e\mathrm {V}$
Forza di Planck	Forza $\left[M\right]\left[L\right]\left[T\right]^{-2}$	$F_{\text{P}}=m_{\text{P}}a_{\text{P}}={\frac {m_{\text{P}}c}{t_{\text{P}}}}={\frac {c^{4}}{4\pi G}}$	${{F}_{\text{P}}}={\frac {{E}_{\text{P}}}{{l}_{\text{P}}}}={\frac {\hbar }{{{l}_{\text{P}}}{{t}_{\text{P}}}}}={\frac {{c}^{4}}{G}}$	$9,630908\cdot 10^{42}\;\mathrm {N}$	$1,210256\cdot 10^{44}\;\mathrm {N}$
Potenza di Planck	Potenza $\left[M\right]\left[L\right]^{2}\left[T\right]^{-3}$	$P_{\text{P}}={\frac {E_{\text{P}}}{t_{\text{P}}}}={\frac {\hbar }{t_{\text{P}}^{2}}}={\frac {c^{5}}{4\pi G}}$	$P_{\text{P}}={\frac {E_{\text{P}}}{t_{\text{P}}}}={\frac {c^{5}}{G}}$	$2,887274\cdot 10^{51}\;\mathrm {W}$	$3,628255\cdot 10^{52}\;\mathrm {W}$
Intensità radiante di Planck	Intensità angolare $\left[L\right]^{2}\left[M\right]\left[T\right]^{-3}$	${\frac {P_{\text{P}}}{\theta _{\text{P}}^{2}}}={\frac {c^{5}}{4\pi G}}$	$\iota _{\text{P}}={\frac {P_{\text{P}}}{\theta _{\text{P}}^{2}}}={\frac {c^{5}}{G}}$	$2,887274\cdot 10^{51}\;{\frac {\mathrm {W} }{\mathrm {sr} }}$	$3,628255\cdot 10^{52}\;{\frac {\mathrm {W} }{\mathrm {sr} }}$
Intensità di Planck	Intensità $\left[M\right]\left[T\right]^{-3}$	$i_{\text{P}}={\frac {P_{\text{P}}}{l_{\text{P}}^{2}}}={\frac {c^{8}}{16\pi ^{2}\hbar G^{2}}}$	$i_{\text{P}}=\rho _{\text{P}}^{E}c={\frac {P_{\text{P}}}{l_{\text{P}}^{2}}}={\frac {c^{8}}{\hbar G^{2}}}$	$8,795455\cdot 10^{119}\;{\frac {\mathrm {W} }{m^{2}}}$	$1,388923\cdot 10^{122}\;{\frac {\mathrm {W} }{m^{2}}}$
Densità di Planck	Densità $\left[M\right]\left[L\right]^{-3}$	$\rho _{\text{P}}={\frac {m_{\text{P}}}{l_{\text{P}}^{3}}}={\frac {\hbar \,t_{\text{P}}}{l_{\text{P}}^{5}}}={\frac {c^{5}}{16\pi ^{2}\hbar G^{2}}}$	$\rho _{\text{P}}={\frac {m_{\text{P}}}{l_{\text{P}}^{3}}}={\frac {c^{5}}{\hbar G^{2}}}$	$3,264346\cdot 10^{94}\;{\frac {kg}{m^{3}}}$	$5,154849\cdot 10^{96}\;{\frac {kg}{m^{3}}}$
Densità energetica di Planck	Densità di energia $\left[L\right]^{-1}\left[M\right]\left[T\right]^{-2}$	$u_{\text{P}}={\frac {E_{\text{P}}}{l_{\text{P}}^{3}}}={\frac {c^{7}}{16\pi ^{2}\hbar G^{2}}}$	$u_{\text{P}}={\frac {E_{\text{P}}}{l_{\text{P}}^{3}}}={\frac {c^{7}}{\hbar G^{2}}}$	$2,933848\cdot 10^{111}\;{\frac {\mathrm {J} }{m^{3}}}$	$4,632947\cdot 10^{113}\;{\frac {\mathrm {J} }{m^{3}}}$
Frequenza angolare di Planck	Frequenza $\left[T\right]^{-1}$	$\omega _{\text{P}}={\frac {\theta _{P}}{t_{\text{P}}}}={\sqrt {\frac {c^{5}}{4\pi \hbar G}}}$	$\omega _{P}={\frac {\theta _{\text{P}}}{t_{\text{P}}}}={\sqrt {\frac {c^{5}}{\hbar G}}}\;$	$5,232458\cdot 10^{42}{\frac {\mathrm {rad} }{s}}$	$1,854858\cdot 10^{43}{\frac {\mathrm {rad} }{s}}$
Accelerazione angolare di Planck	Accelerazione angolare $\left[T\right]^{-2}$	${\frac {\omega _{\text{P}}}{t_{\text{P}}}}=t_{\text{P}}^{-2}={\frac {c^{5}}{4\pi \hbar G}}$	${\frac {\omega _{\text{P}}}{t_{\text{P}}}}=t_{\text{P}}^{-2}={\frac {c^{5}}{\hbar G}}$	$2,737862\cdot 10^{85}\;{\frac {\mathrm {rad} }{s^{2}}}$	$3,440498\cdot 10^{86}\;{\frac {\mathrm {rad} }{s^{2}}}$
Accelerazione di Planck	Accelerazione $\left[L\right]\left[T\right]^{-2}$	$a_{\text{P}}={\frac {v_{\text{P}}}{t_{\text{P}}}}={\sqrt {\frac {c^{7}}{4\pi \hbar G}}}$	$a_{\text{P}}={\frac {c}{t_{\text{P}}}}={\sqrt {\frac {c^{7}}{\hbar G}}}$	$1,568652\cdot 10^{51}\;{\frac {m}{s^{2}}}$	$5,560726\cdot 10^{51}\;{\frac {m}{s^{2}}}$
Momento inerziale di Planck	Momento di inerzia $\left[L\right]^{2}\left[M\right]$	$m_{\text{P}}l_{\text{P}}^{2}={\sqrt {\frac {4\pi \hbar ^{3}G}{c^{5}}}}$	$m_{\text{P}}l_{\text{P}}^{2}={\sqrt {\frac {\hbar ^{3}G}{c^{5}}}}$	$2,01544\cdot 10^{-77}kg\cdot m^{2}$	$5,68546\cdot 10^{-78}kg\cdot m^{2}$
Momento angolare di Planck	Momento angolare $\left[L\right]^{2}\left[M\right]\left[T\right]^{-1}$	$\hbar _{\text{P}}=m_{\text{P}}l_{\text{P}}^{2}\omega _{\text{P}}=l_{\text{P}}m_{\text{P}}c=E_{\text{P}}t_{\text{P}}=\hbar$		$1.054571817\ldots \cdot 10^{-34}\;\mathrm {J} s$
Coppia di Planck	Torque $\left[L\right]^{2}\left[M\right]\left[T\right]^{-2}$	$\tau _{\text{P}}=F_{\text{P}}l_{\text{P}}={\frac {\hbar _{P}}{t_{\text{P}}}}={\sqrt {\frac {\hbar c^{5}}{4\pi G}}}$	$\tau _{\text{P}}=F_{\text{P}}l_{\text{P}}={\frac {\hbar _{\text{P}}}{t_{\text{P}}}}={\sqrt {\frac {\hbar c^{5}}{G}}}$	$5,518004\cdot 10^{8}\mathrm {N} \cdot m$	$1,956081\cdot 10^{9}\mathrm {N} \cdot m$
Pressione di Planck	Pressione $\left[M\right]\left[L\right]^{-1}\left[T\right]^{-2}$	$p_{\text{P}}={\frac {F_{\text{P}}}{l_{\text{P}}^{2}}}={\frac {\hbar }{l_{\text{P}}^{3}t_{\text{P}}}}={\frac {c^{7}}{16\pi ^{2}\hbar G^{2}}}$	$p_{\text{P}}={\frac {F_{\text{P}}}{l_{\text{P}}^{2}}}={\frac {c^{7}}{\hbar G^{2}}}\;$	$2,933848\cdot 10^{111}\mathrm {Pa}$	$4,632947\cdot 10^{113}\mathrm {Pa}$
Tensione superficiale di Planck	Tensione superficiale $\left[M\right]\left[T\right]^{-2}$	${\frac {F_{\text{P}}}{l_{\text{P}}}}={\sqrt {\frac {c^{11}}{64\pi ^{3}\hbar G^{3}}}}$	${\frac {F_{\text{P}}}{l_{\text{P}}}}={\sqrt {\frac {c^{11}}{\hbar G^{3}}}}$	$1,680941\cdot 10^{77}{\frac {\mathrm {N} }{m}}$	$7,488024\cdot 10^{78}{\frac {\mathrm {N} }{m}}$
Forza superficiale universale di Planck	Forza superficiale universale $\left[L\right]^{-1}\left[M\right]\left[T\right]^{-2}$	$p_{\text{P}}={\frac {F_{\text{P}}}{l_{\text{P}}^{2}}}={\frac {c^{7}}{16\pi ^{2}\hbar G^{2}}}$	$p_{\text{P}}={\frac {F_{\text{P}}}{l_{\text{P}}^{2}}}={\frac {c^{7}}{\hbar G^{2}}}$	$2,933848\cdot 10^{111}\mathrm {Pa}$	$4,632947\cdot 10^{113}\mathrm {Pa}$
Durezza di indentazione di Planck	Durezza di indentazione $\left[L\right]^{-1}\left[M\right]\left[T\right]^{-2}$	$p_{\text{P}}={\frac {F_{\text{P}}}{l_{\text{P}}^{2}}}={\frac {c^{7}}{16\pi ^{2}\hbar G^{2}}}$	$p_{\text{P}}={\frac {F_{\text{P}}}{l_{\text{P}}^{2}}}={\frac {c^{7}}{\hbar G^{2}}}$	$2,933848\cdot 10^{111}\mathrm {Pa}$	$4,632947\cdot 10^{113}\mathrm {Pa}$
Durezza assoluta di Planck	Durezza Assoluta $\left[L\right]^{-1}\left[M\right]\left[T\right]^{-2}$	${\frac {a_{\oplus }}{F_{\text{P}}}}={\frac {_{9,80665}\,4\pi G}{c^{4}}}$	${\frac {a_{\oplus }}{F_{\text{P}}}}={\frac {_{9,80665}G}{c^{4}}}$	$1,01825\cdot 10^{-42}kg\cdot f$	$8,10296\cdot 10^{-44}kg\cdot f$
Flusso di massa di Planck	Rapporto di flusso di massa $\left[M\right]\left[T\right]^{-1}$	${t_{r}}_{\text{s}}^{-1}={\frac {m_{\text{P}}}{t_{\text{P}}}}={\frac {c}{2\pi r_{s}}}={\frac {c^{3}}{4\pi G}}$	${t_{r}}_{\text{s}}^{-1}={\frac {m_{\text{P}}}{t_{\text{P}}}}={\frac {2c}{r_{s}}}={\frac {c^{3}}{G}}$	$3,212525\cdot 10^{34}\;{\frac {kg}{s}}$	$4,036978\cdot 10^{35}\;{\frac {kg}{s}}$
Viscosità di Planck	viscosità dinamica $\left[L\right]^{-1}\left[M\right]\left[T\right]^{-1}$	$\eta _{\text{P}}=P_{\text{P}}t_{\text{P}}={\sqrt {\frac {c^{9}}{64\pi ^{3}\hbar G^{3}}}}$	$\eta _{\text{P}}=P_{\text{P}}t_{\text{P}}={\sqrt {\frac {c^{9}}{\hbar G^{3}}}}$	$5,607015\cdot 10^{68}\mathrm {Pa} \cdot s$	$2,497736\cdot 10^{70}\mathrm {Pa} \cdot s$
Viscosità cinematica di Planck	viscosità cinematica $\left[L\right]^{2}\left[T\right]^{-1}$	${\frac {\eta _{\text{P}}}{\rho _{\text{P}}}}={\frac {l_{\text{P}}^{2}}{t_{\text{P}}}}={\sqrt {\frac {4\pi \hbar G}{c}}}$	${\frac {\eta _{\text{P}}}{\rho _{\text{P}}}}={\frac {l_{\text{P}}^{2}}{t_{\text{P}}}}={\sqrt {\frac {\hbar G}{c}}}$	$1,717653\cdot 10^{-27}{\frac {m^{2}}{s}}$	$4,845411\cdot 10^{-27}{\frac {m^{2}}{s}}$
Portata volumetrica di Planck	Rapporto di flusso volumetrico $\left[L\right]^{3}\left[T\right]^{-1}$	$Q_{\text{P}}={\frac {l_{\text{P}}^{3}}{t_{\text{P}}}}=l_{\text{P}}^{2}v_{\text{P}}={\frac {4\pi \hbar G}{c^{2}}}$	$Q_{\text{P}}={\frac {l_{\text{P}}^{3}}{t_{\text{P}}}}=l_{\text{P}}^{2}v_{\text{P}}={\frac {\hbar \,G}{c^{2}}}$	$9,841252\cdot 10^{-61}\;{\frac {m^{3}}{s}}$	$7,831419\cdot 10^{-62}\;{\frac {m^{3}}{s}}$
Proprietà elettromagnetiche
Corrente di Planck	Corrente elettrica $\left[Q\right]\left[T\right]^{-1}$	$I_{\text{P}}={\frac {q_{\text{P}}}{t_{\text{P}}}}={\sqrt {\frac {\varepsilon _{0}c^{6}}{4\pi G}}}$	$I_{\text{P}}={\frac {q_{\text{P}}}{t_{\text{P}}}}={\sqrt {\frac {4\pi \varepsilon _{0}c^{6}}{G}}}$	$2,768399\cdot 10^{24}\;\mathrm {A}$	$3,478873\cdot 10^{25}\;\mathrm {A}$
Forza magnetomotiva di Planck	Corrente elettrica $\left[Q\right]\left[T\right]^{-1}$	$I_{\text{P}}={\frac {q_{\text{P}}}{t_{\text{P}}}}={\sqrt {\frac {\varepsilon _{0}c^{6}}{4\pi G}}}$	$I_{\text{P}}={\frac {q_{\text{P}}}{t_{\text{P}}}}={\sqrt {\frac {4\pi \varepsilon _{0}c^{6}}{G}}}$	$2,768399\cdot 10^{24}\;\mathrm {A}$	$3,478873\cdot 10^{25}\;\mathrm {A}$
Tensione di Planck	Tensione $\left[M\right]\left[L\right]^{2}\left[T\right]^{-2}\left[Q\right]^{-1}$	$V_{\text{P}}={\frac {E_{\text{P}}}{q_{\text{P}}}}={\sqrt {\frac {c^{4}}{4\pi \varepsilon _{0}G}}}$		$1,042940\cdot 10^{27}\;\mathrm {V}$
Forza elettromotiva di Planck	Tensione $\left[M\right]\left[L\right]^{2}\left[T\right]^{-2}\left[Q\right]^{-1}$	$\phi _{\text{P}}=V_{\text{P}}={\frac {E_{\text{P}}}{q_{\text{P}}}}={\sqrt {\frac {c^{4}}{4\pi \varepsilon _{0}G}}}$		$1.042\;940\cdot 10^{27}\;\mathrm {V}$
Resistenza di Planck	Resistenza elettrica $\left[M\right]\left[L\right]^{2}\left[T\right]^{-1}\left[Q\right]^{-2}$	$Z_{\text{P}}={\frac {V_{\text{P}}}{I_{\text{P}}}}={\frac {\hbar }{q_{\text{P}}^{2}}}={\frac {1}{\varepsilon _{0}c}}=\mu _{0}c=Z_{0}$	$Z_{\text{P}}={\frac {V_{\text{P}}}{I_{\text{P}}}}={\frac {1}{4\pi \varepsilon _{0}c}}={\frac {Z_{0}}{4\pi }}$	$376,730\;\Omega$	$29,9792458\;\Omega$
Conduttanza di Planck	Conduttanza elettrica $\left[L\right]^{-2}\left[M\right]^{-1}\left[T\right]\left[Q\right]^{2}$	$G_{\text{P}}={\frac {1}{R_{\text{P}}}}=\varepsilon _{0}c={\frac {1}{Z_{0}}}$	$G_{\text{P}}={\frac {1}{R_{\text{P}}}}=4\pi \varepsilon _{0}c={\frac {4\pi }{Z_{0}}}$	$0,002654\;\mathrm {S}$	$0,0333564095\;\mathrm {S}$
Capacità elettrica di Planck	Capacità elettrica $\left[L\right]^{-2}\left[M\right]^{-1}\left[T\right]^{2}\left[Q\right]^{2}$	${{C}_{\text{P}}}={\frac {{q}_{\text{P}}}{{V}_{\text{P}}}}={\frac {{l}_{P}}{{k}_{e}}}={\sqrt {\frac {4{\pi }\varepsilon _{0}^{2}\hbar G}{{c}^{3}}}}$	${{C}_{\text{P}}}={\frac {{q}_{\text{P}}}{{V}_{\text{P}}}}={\frac {{l}_{P}}{{k}_{e}}}={\sqrt {\frac {16{{\pi }^{2}}\varepsilon _{0}^{2}\hbar G}{{c}^{3}}}}$	$5,072985\cdot 10^{-46}\;\mathrm {F}$	$1,798326\cdot 10^{-45}\;\mathrm {F}$
Permittività di Planck (Costante elettrica)	Permittività elettrica $\left[L\right]^{-3}\left[M\right]^{-1}\left[T\right]^{2}\left[Q\right]^{2}$	$\varepsilon _{\text{P}}={\frac {C_{\text{P}}}{l_{\text{P}}}}={\frac {q_{\text{P}}}{V_{\text{P}}l_{\text{P}}}}={\frac {F_{\text{P}}}{V_{\text{P}}^{2}}}=\varepsilon _{0}$	$\varepsilon _{\text{P}}={\frac {C_{\text{P}}}{l_{\text{P}}}}={\frac {F_{\text{P}}}{V_{\text{P}}^{2}}}={\frac {1}{k_{\text{e}}}}=4\pi \varepsilon _{0}$	$8,854187813\cdot 10^{-12}{\frac {\mathrm {F} }{m}}$	$1,11265006\cdot 10^{-10}{\frac {\mathrm {F} }{m}}$
Permeabilità di Planck (Costante magnetica)	Permeabilità magnetica $\left[L\right]\left[M\right]\left[Q\right]^{-2}$	$\mu _{\text{P}}={\frac {L_{\text{P}}}{l_{\text{P}}}}={\frac {{\phi }_{\text{P}}^{B}}{l_{\text{P}}}}={\frac {1}{\varepsilon _{0}c^{2}}}=\mu _{0}$	$\mu _{\text{P}}={\frac {L_{\text{P}}}{l_{\text{P}}}}={\frac {V_{\text{P}}}{{I_{m}}_{\text{P}}}}={\frac {1}{4\pi \,\varepsilon _{0}c^{2}}}={\frac {\mu _{0}}{4\pi }}$	$1,25663706212{\frac {\mathrm {\mu H} }{m}}$	$10,0000000055{\frac {\mathrm {\mu H} }{m}}$
Induttanza elettrica di Planck	Induttanza $\left[L\right]^{2}\left[M\right]\left[Q\right]^{-2}$	$L_{\text{P}}={\frac {E_{\text{P}}}{I_{\text{P}}}}={\frac {m_{\text{P}}l_{\text{P}}^{2}}{q_{\text{P}}^{2}}}={\sqrt {\frac {4\pi \hbar G}{\varepsilon _{0}^{2}c^{7}}}}$	$L_{\text{P}}={\frac {E_{\text{P}}}{I_{\text{P}}^{2}}}={\frac {m_{\text{P}}l_{\text{P}}^{2}}{q_{\text{P}}^{2}}}={\sqrt {\frac {G\hbar }{16\pi ^{2}\varepsilon _{0}^{2}c^{7}}}}$	$7,199871\cdot 10^{-41}\mathrm {H}$	$1,61625518\cdot 10^{-42}\mathrm {H}$
Resistività elettrica di Planck	Resistività elettrica $\left[L\right]^{3}\left[M\right]\left[T\right]^{-1}\left[Q\right]^{-2}$	$Z_{\text{P}}^{\rho }=Z_{\text{P}}l_{\text{P}}=t_{\text{P}}k_{\text{e}}={\sqrt {\frac {4\pi \hbar G}{\varepsilon _{0}^{2}c^{5}}}}$	$Z_{\text{P}}^{\rho }=Z_{\text{P}}l_{\text{P}}=t_{\text{P}}k_{\text{e}}={\sqrt {\frac {\hbar G}{16\pi ^{2}\varepsilon _{0}^{2}c^{5}}}}$	$2,15847\cdot 10^{-32}\Omega \cdot m$	$4,84541\cdot 10^{-34}\Omega \cdot m$
Conduttività elettrica di Planck	Conduttività elettrica $\left[L\right]^{-3}\left[M\right]^{-1}\left[T\right]\left[Q\right]^{2}$	$\sigma _{\text{P}}={\frac {1}{Z_{\text{P}}^{\rho }}}={\sqrt {\frac {\varepsilon _{0}^{2}c^{5}}{4\pi \hbar G}}}$	$\sigma _{\text{P}}={\frac {1}{Z_{\text{P}}^{\rho }}}={\sqrt {\frac {16\pi ^{2}\varepsilon _{0}^{2}c^{5}}{\hbar G}}}$	$4,632918\cdot 10^{31}{\frac {\mathrm {S} }{m}}$	$2,063809\cdot 10^{33}{\frac {\mathrm {S} }{m}}$
Densità di carica di Planck	Densità di carica $\left[L\right]^{-3}\left[Q\right]$	${\rho _{e}}_{\text{P}}={\frac {q_{\text{P}}}{l_{\text{P}}^{3}}}={\sqrt {\frac {\varepsilon _{0}c^{10}}{64\pi ^{3}\hbar ^{2}G^{3}}}}$	${\rho _{e}}_{\text{P}}={\frac {q_{\text{P}}}{l_{\text{P}}^{3}}}={\sqrt {\frac {4\pi \varepsilon _{0}c^{10}}{\hbar ^{2}G^{3}}}}$	$2,813056\cdot 10^{86}{\frac {\mathrm {C} }{m^{3}}}$	$4,442200\cdot 10^{86}{\frac {\mathrm {C} }{m^{3}}}$
Forza del campo elettrico di Planck	Campo elettrico $\left[L\right]\left[M\right]\left[T\right]^{-2}\left[Q\right]^{-1}$	${\bf {E}}_{\text{P}}={\frac {F_{\text{P}}}{q_{\text{P}}}}={\sqrt {\frac {c^{7}}{16\pi ^{2}\varepsilon _{0}\hbar G^{2}}}}$	${\bf {E}}_{\text{P}}={\frac {F_{\text{P}}}{q_{\text{P}}}}={\sqrt {\frac {c^{7}}{4\pi \varepsilon _{0}\hbar G^{2}}}}$	$1,820306\cdot 10^{61}{\frac {\mathrm {V} }{m}}$	$6,452817\cdot 10^{61}{\frac {\mathrm {V} }{m}}$
Forza del campo magnetico di Planck	Campo magnetico $\left[L\right]^{-1}\left[T\right]^{-1}\left[Q\right]$	${\bf {H}}_{\text{P}}={\frac {I_{\text{P}}}{l_{\text{P}}}}={\sqrt {\frac {\varepsilon _{0}c^{9}}{16\pi ^{2}\hbar G^{2}}}}$	${\bf {H}}_{\text{P}}={\frac {I_{\text{P}}}{l_{\text{P}}}}={\sqrt {\frac {4\pi \varepsilon _{0}c^{9}}{\hbar G^{2}}}}$	$4,831855\cdot 10^{58}{\frac {\mathrm {A} }{m}}$	$2,152428\cdot 10^{60}{\frac {\mathrm {A} }{m}}$
Induzione elettrica di Planck	Corrente di spostamento $\left[L\right]^{-2}\left[T\right]^{-1}\left[Q\right]$	${\bf {D}}_{\text{P}}={\frac {q_{\text{P}}}{l_{\text{P}}^{2}}}={\sqrt {\frac {\varepsilon _{0}c^{7}}{16\pi ^{2}\hbar G^{2}}}}$	${\bf {D}}_{\text{P}}={\frac {q_{\text{P}}}{l_{\text{P}}^{2}}}={\sqrt {\frac {4\pi \varepsilon _{0}c^{7}}{\hbar G^{2}}}}$	$1,611733\cdot 10^{50}{\frac {\mathrm {C} }{m^{2}}}$	$7,179727\cdot 10^{51}{\frac {\mathrm {C} }{m^{2}}}$
Induzione magnetica di Planck	Campo magnetico $\left[M\right]\left[T\right]^{-1}\left[Q\right]^{-1}$	${\bf {B}}_{\text{P}}={\frac {F_{\text{P}}}{l_{\text{P}}I_{\text{P}}}}={\sqrt {\frac {c^{5}}{16\pi ^{2}\varepsilon _{0}\hbar G^{2}}}}$	${\bf {B}}_{\text{P}}={\frac {F_{\text{P}}}{l_{\text{P}}I_{\text{P}}}}={\frac {\hbar }{q_{\text{P}}l_{\text{P}}^{2}}}={\sqrt {\frac {c^{5}}{4\pi \varepsilon _{0}\hbar G^{2}}}}$	$6,071888\cdot 10^{52}\;\mathrm {T}$	$2,152428\cdot 10^{53}\;\mathrm {T}$
Flusso elettrico di Planck	Flusso magnetico $\left[L\right]^{2}\left[M\right]\left[T\right]^{-1}\left[Q\right]^{-1}$	${\phi }_{\text{P}}^{E}={\bf {E}}_{\text{P}}l_{\text{P}}^{2}=\phi _{\text{P}}l_{\text{P}}={\sqrt {\frac {\hbar c}{\varepsilon _{0}}}}$	${\phi }_{\text{P}}^{E}={\bf {E}}_{\text{P}}l_{\text{P}}^{2}=\phi _{\text{P}}l_{\text{P}}={\sqrt {\frac {\hbar c}{4\pi \varepsilon _{0}}}}$	$5,975498\cdot 10^{-8}\mathrm {V} \cdot m$	$1,685657\cdot 10^{-8}\mathrm {V} \cdot m$
Flusso magnetico di Planck	Flusso magnetico $\left[L\right]^{2}\left[M\right]\left[T\right]^{-1}\left[Q\right]^{-1}$	${\phi }_{\text{P}}^{B}={\bf {B}}_{\text{P}}l_{\text{P}}^{2}={\bf {A}}_{\text{P}}l_{\text{P}}={\sqrt {\frac {\hbar }{\varepsilon _{0}c}}}$	${\phi }_{\text{P}}^{B}={\frac {E_{\text{P}}}{I_{\text{P}}}}={\bf {A}}_{\text{P}}l_{\text{P}}={\frac {\hbar }{q_{\text{P}}}}={\sqrt {\frac {\hbar }{4\pi \varepsilon _{0}c}}}$	$1,993211\cdot 10^{-16}\,\mathrm {Wb}$	$5,622746\cdot 10^{-17}\;\mathrm {Wb}$
Potenziale elettrico di Planck	Tensione $\left[L\right]^{2}\left[M\right]\left[T\right]^{-2}\left[Q\right]^{-1}$	$\phi _{\text{P}}=V_{\text{P}}={\frac {E_{\text{P}}}{q_{\text{P}}}}={\sqrt {\frac {c^{4}}{4\pi \varepsilon _{0}G}}}$		$1,042940\cdot 10^{27}\;\mathrm {V}$
Potenziale magnetico di Planck	Corrente magnetica $\left[L\right]\left[M\right]\left[T\right]^{-1}\left[Q\right]^{-1}$	${{\bf {A}}_{\text{P}}}={\frac {{E}_{\text{P}}}{{{q}_{m}}_{\text{P}}}}={\frac {{F}_{\text{P}}}{{I}_{\text{P}}}}={\frac {{V}_{\text{P}}}{{v}_{\text{P}}}}={{\bf {B}}_{\text{P}}}{{l}_{\text{P}}}={\frac {\hbar }{{{q}_{\text{P}}}{{l}_{\text{P}}}}}={\sqrt {\frac {{c}^{2}}{4\pi {{\varepsilon }_{0}}G}}}$		$3,478873\cdot 10^{18}\;\mathrm {T} \cdot m$
Densità di corrente di Planck	Densità di corrente elettrica $\left[L\right]^{-2}\left[T\right]^{-1}\left[Q\right]$	${\bf {J}}_{\text{P}}={\frac {I_{\text{P}}}{l_{\text{P}}^{2}}}={{\rho }_{e}}_{\text{P}}v_{\text{P}}={\sqrt {\frac {\varepsilon _{0}c^{12}}{64\pi ^{3}\hbar ^{2}G^{3}}}}$	${\bf {J}}_{\text{P}}={\frac {I_{\text{P}}}{l_{\text{P}}^{2}}}={{\rho }_{e}}_{\text{P}}v_{\text{P}}={\sqrt {\frac {4\pi \varepsilon _{0}c^{12}}{\hbar ^{2}G^{3}}}}$	$8,433329\cdot 10^{92}\;{\frac {\mathrm {A} }{m^{2}}}$	$1,331738\cdot 10^{95}\;{\frac {\mathrm {A} }{m^{2}}}$
Momento elettrico di Planck	Dipolo elettrico $\left[L\right]\left[Q\right]$	${d}_{\text{P}}=q_{\text{P}}l_{\text{P}}={\sqrt {\frac {4\pi \varepsilon _{0}\hbar ^{2}G}{c^{2}}}}$		$3,031361\cdot 10^{-53}\;\mathrm {C} \cdot m$
Momento magnetico di Planck	Dipolo magnetico $\left[L\right]^{2}\left[T\right]^{-1}\left[Q\right]$	${\mu _{d}}_{\text{P}}={q_{m}}_{\text{P}}l_{\text{P}}=I_{\text{P}}l_{\text{P}}^{2}={\sqrt {4\pi \varepsilon _{0}\hbar ^{2}G}}$		$9,087791\cdot 10^{-45}\;{\frac {\mathrm {J} }{\mathrm {T} }}$
Monopolo magnetico di Planck	Carica magnetica $\left[L\right]\left[T\right]^{-1}\left[Q\right]$	${{q}_{m}}_{\text{P}}=q_{\text{P}}v_{P}={\frac {F_{\text{P}}}{{\bf {B}}_{\text{P}}}}={\sqrt {{{\varepsilon }_{0}}\hbar {{c}^{3}}}}$	${{q}_{m}}_{\text{P}}=q_{\text{P}}v_{P}={\frac {4\pi }{\mu _{0}}}\phi _{\text{P}}^{B}={\sqrt {4\pi {{\varepsilon }_{0}}\hbar {{c}^{3}}}}$	$1.586147\cdot 10^{-10}{\frac {\mathrm {N} }{\mathrm {T} }}$	$5.622746\cdot 10^{-10}\mathrm {A} \cdot m$
Corrente magnetica di Planck	Corrente magnetica $\left[L\right]\left[T\right]^{-2}\left[Q\right]$	${I_{m}}_{\text{P}}={\frac {{q_{m}}_{\text{P}}}{t_{\text{P}}}}={q_{\text{P}}}{a_{\text{P}}}={I_{\text{P}}}{v_{\text{P}}}={\sqrt {\frac {\varepsilon _{0}c^{8}}{4\pi G}}}$	${I_{m}}_{\text{P}}={\frac {{q_{m}}_{\text{P}}}{t_{\text{P}}}}={q_{\text{P}}}{a_{\text{P}}}={\sqrt {\frac {4\pi \varepsilon _{0}c^{8}}{G}}}$	$8,29945\cdot 10^{32}{\frac {\mathrm {V} \cdot m}{\mathrm {H} }}$	$1,04294\cdot 10^{34}{\frac {\mathrm {W} }{\mathrm {T} \cdot m}}$
Densità di corrente magnetica di Planck	Corrente magnetica $\left[L\right]^{-1}\left[T\right]^{-2}\left[Q\right]$	${\frac {{I_{m}}_{\text{P}}}{l_{\text{P}}^{2}}}={\bf {J}}_{\text{P}}v_{\text{P}}={\frac {{I}_{\text{P}}}{l_{\text{P}}t_{\text{P}}}}={\sqrt {\frac {\varepsilon _{0}c^{14}}{64\pi ^{3}\hbar ^{2}G^{3}}}}$	${\frac {{I_{m}}_{\text{P}}}{l_{\text{P}}^{2}}}={\bf {J}}_{\text{P}}v_{\text{P}}={\frac {{I}_{\text{P}}}{l_{\text{P}}t_{\text{P}}}}={\sqrt {\frac {4\pi \varepsilon _{0}c^{14}}{\hbar ^{2}G^{3}}}}$	$2,52825\cdot 10^{101}{\frac {\mathrm {V} }{m\cdot \mathrm {H} }}$	$3,99245\cdot 10^{103}{\frac {\mathrm {V} }{m\cdot \mathrm {H} }}$
Carica specifica di Planck	carica specifica $\left[M\right]^{-1}\left[Q\right]$	$q_{r_{\text{s}}}={\frac {q_{\text{P}}}{m_{\text{P}}}}={\sqrt {\frac {2\pi {r_{\text{s}}}}{\mu _{0}}}}={\sqrt {\frac {G}{k_{e}}}}={\sqrt {4\pi \varepsilon _{0}G}}$		$8.617517\cdot 10^{-11}\;{\frac {\mathrm {Hz} }{\mathrm {T} }}$
Monopolo specifica di Planck^{[non chiaro]}	carica magnetica specifica $\left[L\right]\left[T\right]^{-1}\left[M\right]^{-1}\left[Q\right]$	$q_{r_{\text{s}}}c={\frac {q_{\text{P}}c}{m_{\text{P}}}}={\frac {a_{\text{P}}}{{\bf {B}}_{\text{P}}}}={\sqrt {4\pi \varepsilon _{0}c^{2}G}}$	$q_{r_{\text{s}}}c={\frac {q_{\text{P}}c}{m_{\text{P}}}}={\frac {a_{\text{P}}}{{\bf {B}}_{\text{P}}}}={\sqrt {\frac {4\pi G}{\mu _{0}}}}$	$0,0258347{\frac {m}{s^{2}\cdot \mathrm {T} }}$	$0,0258347{\frac {m}{s^{2}\cdot \mathrm {T} }}$
Proprietà termodinamiche
Temperatura di Planck in 2π	Temperatura $\left[\Theta \right]$	${\Theta }_{\text{P}}^{_{2\pi }}=2\pi {\Theta _{\text{P}}}={\frac {2\pi E_{\text{P}}}{k_{\text{B}}}}={\sqrt {\frac {\pi \hbar c^{5}}{G{k_{\text{B}}^{2}}}}}$	$\Theta _{\text{P}}^{_{2\pi }}=2\pi {\Theta _{\text{P}}}={\frac {2\pi m_{\text{P}}c^{2}}{k_{\text{B}}}}={\sqrt {\frac {\pi \hbar c^{5}}{Gk_{\text{B}}^{2}}}}$	$2,511185\cdot 10^{32}\mathrm {K}$	$8,901917\cdot 10^{32}\mathrm {K}$
Entropia di Planck	Entropia $\left[L\right]^{2}\left[M\right]\left[T\right]^{-2}\left[\Theta \right]^{-1}$	$S_{\text{P}}={\frac {E_{\text{P}}}{\Theta _{\text{P}}}}=k_{\text{B}}$		$1,380649\cdot 10^{-23}{\frac {\mathrm {J} }{\mathrm {K} }}$
Entropia di Planck in 2 π	Entropia $\left[L\right]^{2}\left[M\right]\left[T\right]^{-2}\left[\Theta \right]^{-1}$	${S_{2\pi }}_{\text{P}}={\frac {E_{\text{P}}}{2\pi \Theta _{\text{P}}}}={\frac {k_{\text{B}}}{2\pi }}$		$2,197371\cdot 10^{-24}{\frac {\mathrm {J} }{\mathrm {K} }}$
Coefficiente di dilatazione termica di Planck	Coefficiente di dilatazione termica $\left[\Theta \right]^{-1}$	${\alpha _{_{V}}}_{\text{P}}={\frac {1}{\Theta _{\text{P}}}}={\frac {k_{\text{B}}}{E_{\text{P}}}}={\sqrt {\frac {4\pi G{k_{\text{B}}}^{2}}{\hbar c^{5}}}}$	${\alpha _{_{V}}}_{\text{P}}={\frac {1}{\Theta _{\text{P}}}}={\frac {k_{\text{B}}}{E_{\text{P}}}}={\sqrt {\frac {G{k_{\text{B}}}^{2}}{\hbar c^{5}}}}$	$2,502080\cdot 10^{-33}{\frac {1}{\mathrm {K} }}$	$7,058238\cdot 10^{-33}{\frac {1}{\mathrm {K} }}$
Capacità termica di Planck	Capacità termica - Entropia $\left[L\right]^{2}\left[M\right]\left[T\right]^{-2}\left[\Theta \right]^{-1}$	${C}_{\text{P}}^{\Theta }={\frac {E_{\text{P}}}{\Theta _{\text{P}}}}=k_{\text{B}}$		$1,380649\cdot 10^{-23}{\frac {\mathrm {J} }{\mathrm {K} }}$
Calore specifico di Planck	Calore specifico $\left[L\right]^{2}\left[T\right]^{-2}\left[\Theta \right]^{-1}$	${c_{p}}_{\text{P}}={\frac {E_{\text{P}}}{m_{\text{P}}\Theta _{\text{P}}}}={\frac {k_{\text{B}}}{m_{\text{P}}}}={\sqrt {\frac {4\pi Gk_{\text{B}}^{2}}{\hbar c}}}$	${c_{p}}_{\text{P}}={\frac {E_{\text{P}}}{m_{\text{P}}\Theta _{\text{P}}}}={\frac {k_{\text{B}}}{m_{\text{P}}}}={\sqrt {\frac {Gk_{\text{B}}^{2}}{\hbar c}}}$	$2,24876\cdot 10^{-15}{\frac {\mathrm {J} }{kg\cdot \mathrm {K} }}$	$6,34363\cdot 10^{-16}{\frac {\mathrm {J} }{kg\cdot \mathrm {K} }}$
Calore volumetrico di Planck	Calore volumetrico $\left[L\right]^{-1}\left[M\right]\left[T\right]^{-2}\left[\Theta \right]^{-1}$	${c_{V}}_{\text{P}}={\frac {E_{\text{P}}}{l_{\text{P}}^{3}\Theta _{\text{P}}}}={\frac {k_{\text{B}}}{l_{\text{P}}^{3}}}={\sqrt {\frac {c^{9}k_{\text{B}}^{2}}{64\pi ^{3}\hbar ^{3}G^{3}}}}$	${c_{V}}_{\text{P}}={\frac {E_{\text{P}}}{l_{\text{P}}^{3}\Theta _{\text{P}}}}={\frac {k_{\text{B}}}{l_{\text{P}}^{3}}}={\sqrt {\frac {c^{9}k_{\text{B}}^{2}}{\hbar ^{3}G^{3}}}}$	$7,340723\cdot 10^{79}{\frac {\mathrm {J} }{m^{3}\cdot \mathrm {K} }}$	$3,270044\cdot 10^{81}{\frac {\mathrm {J} }{m^{3}\cdot \mathrm {K} }}$
Resistenza termica di Planck	Resistenza termica $\left[L\right]^{-2}\left[M\right]^{-1}\left[T\right]^{3}\left[\Theta \right]$	${\Omega _{\Theta }}_{\text{P}}={\frac {\Theta _{\text{P}}}{P_{\text{P}}}}={\frac {t_{\text{P}}}{k_{\text{B}}}}={\sqrt {\frac {4\pi \hbar G}{c^{5}k_{\text{B}}^{2}}}}$	${\Omega _{\Theta }}_{\text{P}}={\frac {\Theta _{\text{P}}}{P_{\text{P}}}}={\frac {t_{\text{P}}}{k_{\text{B}}}}={\sqrt {\frac {\hbar G}{c^{5}k_{\text{B}}^{2}}}}$	$1,384238\cdot 10^{-20}{\frac {\mathrm {K} }{\mathrm {W} }}$	$3,904864\cdot 10^{-21}{\frac {\mathrm {K} }{\mathrm {W} }}$
Conduttanza termica di Planck	Conduttanza termica $\left[L\right]^{2}\left[M\right]\left[T\right]^{-3}\left[\Theta \right]^{-1}$	${G_{\Theta }}_{\text{P}}={\frac {k_{\text{B}}}{t_{\text{P}}}}={\sqrt {\frac {c^{5}k_{\text{B}}^{2}}{4\pi \hbar G}}}\simeq {\bf {A}}_{\text{P}}{\frac {2\pi }{\sqrt {\alpha }}}$	${G_{\Theta }}_{\text{P}}={\frac {1}{{\Omega _{\Theta }}_{\text{P}}}}={\frac {k_{\text{B}}}{t_{\text{P}}}}={\sqrt {\frac {c^{5}k_{\text{B}}^{2}}{\hbar G}}}$	$7,224190\cdot 10^{19}{\frac {\mathrm {W} }{\mathrm {K} }}$	$2,560909\cdot 10^{20}{\frac {\mathrm {W} }{\mathrm {K} }}$
Resistività termica di Planck	Resistività termica $\left[L\right]^{-1}\left[M\right]^{-1}\left[T\right]^{3}\left[\Theta \right]$	${\frac {1}{{\lambda _{\Theta }}_{\text{P}}}}={\Omega _{\Theta }}_{\text{P}}l_{\text{P}}={\frac {l_{\text{P}}t_{\text{P}}}{k_{\text{B}}}}={\sqrt {\frac {16\pi ^{2}\hbar ^{2}G^{2}}{c^{8}k_{\text{B}}^{2}}}}$

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Unità di misura di Planck derivate

Tabella[modifica | modifica wikitesto]

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