College General Astronomy
EQUATIONS
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INFORMATION
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$$ \large a = \frac{r + r' }{2} $$
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\( \textbf{a} = semi - major \hspace{4 pt} axis \hspace{4 pt} (m) \\ \textbf{r} = distance \hspace{4 pt} between \hspace{4 pt} focus \\ \hspace{10 pt} and \hspace{4 pt} object \hspace{4 pt} (m) \\ \textbf{r'} = distance \hspace{4 pt} between \hspace{4 pt} focus' \\ \hspace{10 pt} and \hspace{4 pt} object \hspace{4 pt} (m) \) |
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$$ \large r( \theta ) = \frac{a (1-e^2)}{1 + e \hspace{2 pt} cos(\theta)} $$
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\( r( \theta ) = radius \hspace{4 pt} of \hspace{4 pt} orbit \hspace{4 pt} (m) \\ \textbf{a} = semi - major \hspace{4 pt} axis \hspace{4 pt} (m) \\ \textbf{e} = eccentricity \\ \theta = angular \hspace{4 pt} coordinate \hspace{4 pt} (deg) \) |
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$$ \large \\
\theta = \frac{S}{r} \\ \\
\Omega = \frac{A}{r^2} $$
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\(\\ \theta = angle \hspace{4 pt} subtended \hspace{4 pt} by \hspace{4 pt} arc \hspace{4 pt} S \hspace{4 pt} (deg) \\ \textbf{S} = arc \hspace{4 pt} length \hspace{4 pt} (m) \\ \textbf{r} = radius \hspace{4 pt} of \hspace{4 pt} circle \hspace{4 pt} (m) \\ \Omega = solid \hspace{4 pt} angle \hspace{4 pt} subtended \hspace{4 pt} by \\ \hspace{4 pt} surface \hspace{4 pt} area \hspace{4 pt} A \hspace{4 pt} (sr) \\ \textbf{A} = surface \hspace{4 pt} area \hspace{4 pt} (m)\) |
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$$ \large \frac{1}{P} = \frac{1}{E} - \frac{1}{S} $$
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\( \textbf{P} = inferior \hspace{4 pt} planet's \hspace{4 pt} sidereal \hspace{4 pt} period \hspace{4 pt} (yr) \\ \textbf{E} = Earth's \hspace{4 pt} sidereal \hspace{4 pt} period \hspace{4 pt} (1 \hspace{2 pt} yr) \) |
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$$ \large \frac{1}{P} = \frac{1}{E} + \frac{1}{S} $$
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\( \textbf{P} = inferior \hspace{4 pt} planet's \hspace{4 pt} sidereal \hspace{4 pt} period \hspace{4 pt} (yr) \\ \textbf{E} = Earth's \hspace{4 pt} sidereal \hspace{4 pt} period \hspace{4 pt} (1 \hspace{2 pt} yr) \) |
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$$ \large {F} = ma $$
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\( \textbf{F} = force \hspace{4 pt} on \hspace{4 pt} an \hspace{4 pt} object \hspace{4 pt} (N) \\ \textbf{m} = mass \hspace{4 pt} of \hspace{4 pt} the \hspace{4 pt} object \hspace{4 pt} (kg) \\ \textbf{a} = acceleration \hspace{4 pt} of \hspace{4 pt} object \hspace{4 pt} (m \hspace{2 pt} s^{-2}) \) |
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$$ \large
F = G \left( \frac{m_1 m_2}{r^2} \right ) $$
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\( \textbf{F} = gravitational \hspace{4 pt} force \hspace{4 pt} between \\ \hspace{6 pt} masses \hspace{4 pt} (N)\\ \textbf{G} = gravitational \hspace{4 pt} constant \\ \hspace{7 pt} (6.67384 x 10^{-11} \hspace{2 pt} m^3 \hspace{2 pt} kg^{-1} \hspace{2 pt} s^{-2}) \\ {m_\#} = mass \hspace{4 pt} of \hspace{4 pt} object \hspace{4 pt} \# \hspace{4 pt} (kg)\\ \textbf{r} = distance \hspace{4 pt} between \hspace{4 pt} objects \hspace{4 pt} (m)\) |
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$$ \large F_{tidal} = \frac{2GM_{Earth} m d}{r^3} $$
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\( {F_{tidal}} = tidal \hspace{4 pt} force \hspace{4 pt} (N) \\ \textbf{G} = gravitational \hspace{4 pt} constant \\ \hspace{7 pt} (6.67384 x 10^{-11} \hspace{2 pt} m^3 \hspace{2 pt} kg^{-1} \hspace{2 pt} s^{-2}) \\ {M_{Earth}} = mass \hspace{4 pt} of \hspace{4 pt} the \hspace{4 pt} Earth \\ \hspace{15 pt} (5.974 \hspace{2 pt} x \hspace{2 pt} 10^{24} \hspace{2 pt} kg) \\ \textbf{m} = mass \hspace{4 pt} of \hspace{4 pt} object \hspace{4 pt} experiencing \\ \hspace{8 pt} force \hspace{4 pt} (kg) \\ \textbf{d} = Moon's \hspace{4 pt} diameter \hspace{4 pt} (3.476 \hspace{2 pt} x \hspace{2 pt} 10^6 \hspace{2 pt} m)\\ \textbf{r} = Earth-Moon \hspace{4 pt} distance \\ \hspace{4 pt} (3.844 \hspace{2 pt} x \hspace{2 pt} 10^8 \hspace{2 pt} m)\) |
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$$ \large
T^2 = \left ( \frac{4\pi^2}{G(m_1 + m_2)} \right ) a^3 $$
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\(\\ \textbf{T} = orbital \hspace{4 pt} period \hspace{4 pt} (s) \\ \textbf{G} = gravitational \hspace{4 pt} constant \hspace{7 pt} \\ \hspace{8 pt} (6.67384 x 10^{-11} \hspace{2 pt} m^3 \hspace{2 pt} kg^{-1} \hspace{2 pt} s^{-2}) \\ {m_\#} = mass \hspace{4 pt} of \hspace{4 pt} object \hspace{4 pt} \# \hspace{4 pt} (kg)\\ \textbf{a} = semi-major \hspace{4 pt} axis \hspace{4 pt} (m)\) |
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$$ \large L = mvr $$
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\(\\ \textbf{L} = angular \hspace{4 pt} momentum \hspace{4 pt} (kg \hspace{2 pt} m^2 \hspace{2 pt} s^{-1}) \\ \textbf{m} = mass \hspace{4 pt} (kg) \\ \textbf{v} = velocity \hspace{4 pt} (m \hspace{2 pt} s^{-1}) \\ \textbf{r} = distance \hspace{4 pt} from \hspace{4 pt} point \hspace{4 pt} of \hspace{4 pt} rotation \hspace{4 pt} (m)\) |
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$$ \large
U = -\frac{GMm}{r} $$
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\(\\ \textbf{U} = gravitational \hspace{4 pt} potential \hspace{4 pt} energy (J) \\ \textbf{G} = gravitational \hspace{4 pt} constant \\ \hspace{8 pt} (6.67384 x 10^{-11} \hspace{2 pt} m^3 \hspace{2 pt} kg^{-1} \hspace{2 pt} s^{-2}) \\ \textbf{M} = mass \hspace{4 pt} of \hspace{4 pt} planet \hspace{4 pt} (kg) \\ \textbf{m} = mass \hspace{4 pt} of \hspace{4 pt} object \hspace{4 pt} (kg) \\ \textbf{r} = distance \hspace{4 pt} between \hspace{4 pt} the \hspace{4 pt} center \\ \hspace{8 pt} of \hspace{4 pt} masses \hspace{4 pt} of \hspace{4 pt} the \hspace{4 pt} objects \hspace{4 pt} (m)\) |
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$$ \large
K = \frac{1}{2} m v^2 = \frac{3}{2}kT $$
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\(\\ \textbf{K} = kinetic \hspace{4 pt} energy \hspace{4 pt} (J) \\ \textbf{m} = mass \hspace{4 pt} (kg) \\ \textbf{v} = velocity \hspace{4 pt} (m \hspace{2 pt} s^{-1}) \\ \textbf{k} = Boltzmann \hspace{4 pt} constant \\ \hspace{5 pt} (1.38 \hspace{2 pt} x 10^{-23} \hspace{2 pt} m^2 \hspace{2 pt} kg \hspace{2 pt} s^{-2} \hspace{2 pt} K^{-1}) \\ \textbf{T} = temperature \hspace{2 pt} (K)\) |
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$$ \large v_{orb} = \sqrt{\frac{GM}{r}} $$
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\(\\ {v_{orb}} = escape \hspace{4 pt} velocity \hspace{4 pt} (m \hspace{2 pt} s^{-1}) \\ \textbf{G} = gravitational \hspace{4 pt} constant \hspace{4 pt} \\ \hspace{8 pt} (6.67 \hspace{2 pt} x \hspace{2 pt} 10^{-11} \hspace{2 pt} m^3 \hspace{2 pt} kg^{-1} \hspace{2 pt} s^{-2}) \\ \textbf{M} = mass \hspace{4 pt} of \hspace{4 pt} planet \hspace{4 pt} (kg) \\ \textbf{r} = distance \hspace{4 pt} from \hspace{4 pt} center \hspace{4 pt} of \\ \hspace{3 pt} gravity \hspace{4 pt} (m)\) |
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$$ \large v_{esc} = \sqrt{\frac{2GM}{r}} $$
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\(\\ {v_{esc}} = escape \hspace{4 pt} velocity \hspace{4 pt} (m \hspace{2 pt} s^{-1}) \\ \textbf{G} = gravitational \hspace{4 pt} constant \hspace{4 pt} \\ \hspace{8 pt} (6.67 \hspace{2 pt} x \hspace{2 pt} 10^{-11} \hspace{2 pt} m^3 \hspace{2 pt} kg^{-1} \hspace{2 pt} s^{-2}) \\ \textbf{M} = mass \hspace{4 pt} of \hspace{4 pt} planet \hspace{4 pt} (kg) \\ \textbf{r} = distance \hspace{4 pt} from \hspace{4 pt} center \hspace{4 pt} of \\ \hspace{3 pt} gravity \hspace{4 pt} (m)\) |
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$$ \large
c = \nu \lambda $$
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\(\\ \textbf{c} = speed \hspace{4 pt} of \hspace{4 pt} light \hspace{4 pt} (3.00 \hspace{2 pt} m \hspace{2 pt} s^{-2}) \\ {\nu} = frequency \hspace{4 pt} of \hspace{4 pt} light \hspace{4 pt} (Hz) \\ {\lambda} = wavelength \hspace{4 pt} of \hspace{4 pt} light \hspace{4 pt} (m)\) |
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$$ \large
\frac{v_r}{c} = \frac{\Delta \lambda}{\lambda_0} = \frac{\lambda - \lambda_0}{\lambda_0} $$
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\(\\ {v_r} = radial \hspace{4 pt} velocity \hspace {4 pt}(km/s) \\ \textbf{c} = speed \hspace {4 pt}of \hspace {4 pt} light \hspace{4 pt}(km/s) \\ {\lambda} = observed \hspace {4 pt} wavelength \hspace {4 pt}(km/s) \\ {\lambda_0} = emitted \hspace {4 pt} wavelength \hspace{4 pt}(km/s) \\ \) |
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$$ \large
E = h\nu = \frac{hc}{\lambda} $$
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\( \textbf{E} = energy \hspace{4 pt} of \hspace{4 pt} a \hspace{4 pt} photon \hspace{4 pt} (J) \\ \textbf{h} = Planck's \hspace{4 pt} constant \hspace{4 pt} (6.625 \hspace{2 pt}x \hspace{2 pt} 10^{-34} J \hspace{2 pt} s ) \\ {\nu} = frequency \hspace{4 pt} of \hspace{4 pt} the \hspace{4 pt} light \hspace{4 pt}(Hz) \\ \textbf{c} = speed \hspace{4 pt} of \hspace{4 pt} light \hspace{4 pt} (3.0 \hspace{2 pt} x \hspace{2 pt} 10^8 \hspace{4 pt} m \hspace{2 pt} s^{-1}) \\ {\lambda} = wavelength \hspace{4 pt} (m) \) |
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$$ \large
h = p \lambda = \frac{E}{\nu} $$
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\(\\ \textbf{E} = energy \hspace{4 pt} of \hspace{4 pt} a \hspace{4 pt} photon \hspace{4 pt} (J) \\ \textbf{h} = Planck's \hspace{4 pt} constant \hspace{4 pt} (6.625 \hspace{2 pt}x \hspace{2 pt} 10^{-34} J \hspace{2 pt} s ) \\ {\nu} = frequency \hspace{4 pt} of \hspace{4 pt} the \hspace{4 pt} light \hspace{4 pt}(Hz) \\ \textbf{p} = momentum \hspace{4 pt} (kg \hspace{2 pt} m \hspace{2 pt} s^{-1}) \\ {\lambda} = wavelength \hspace{4 pt} (m)\) |
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$$ \large
\frac{1}{\lambda} = R \left(\frac{1}{4} - \frac{1}{n^2} \right ) $$
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\(\\ {\lambda} = wavelength \hspace{4 pt} (m) \\ \textbf{R} = Rydberg \hspace{4 pt} constant \hspace{4 pt} \\ \hspace{8 pt}(1.097 \hspace{2 pt} x \hspace{2 pt} 10^7 \hspace{2 pt} m^{-1}) \\ \textbf{n} = number \hspace{4 pt} of \hspace{4 pt} outer \hspace{4 pt} orbit \) |
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$$ \large
\frac{1}{\lambda} = R \left(\frac{1}{N^2} - \frac{1}{n^2} \right ) $$
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\(\\ {\lambda} = wavelength \hspace{4 pt} (m) \\ \textbf{R} = Rydberg \hspace{4 pt} constant \hspace{4 pt} (1.097 \hspace{2 pt} x \hspace{2 pt} 10^7 \hspace{2 pt} m^{-1}) \\ \textbf{N} = Number \hspace{4 pt} of \hspace{4 pt} inner \hspace{4 pt} orbit \\ \textbf{n} = number \hspace{4 pt} of \hspace{4 pt} outer \hspace{4 pt} orbit \) |
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$$ \large dsin(\theta) = n \lambda \hspace{4 pt} (n = 0,1,2,...) $$
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\(\\ \textbf{d} = distance \hspace{4 pt} between \hspace{4 pt} slits \hspace{4 pt} (cm) \\ {\theta} = angle \hspace{4 pt} between \hspace{4 pt} fringes \hspace{4 pt} (deg) \\ \textbf{n} = order \hspace{4 pt} of \hspace{4 pt} the \hspace{4 pt} fringe \\ {\lambda} = wavelength \hspace{4 pt} of \hspace{4 pt} light \hspace{4 pt} (cm)\) |
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$$ \large dsin(\theta) = \left(n - \frac{1}{2} \right) \lambda \hspace{4 pt} (n = 0,1,2,...) $$
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\(\\ \textbf{d} = distance \hspace{4 pt} between \hspace{4 pt} slits \hspace{4 pt} (cm) \\ {\theta} = angle \hspace{4 pt} between \hspace{4 pt} fringes \hspace{4 pt} (deg) \\ \textbf{n} = order \hspace{4 pt} of \hspace{4 pt} the \hspace{4 pt} fringe \\ {\lambda} = wavelength \hspace{4 pt} of \hspace{4 pt} light \hspace{4 pt} (cm)\) |
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$$ \large B(\lambda,T) = \frac{2hc^2}{\lambda^5(e^{\frac{hc}{\lambda k T}}-1)} $$
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\( \textbf{B} = spectral \hspace{4 pt} radiance \hspace{4 pt} (W \hspace{2 pt} sr^{-1} \hspace{2 pt} m^{-3}) \\ \textbf{h} = Planck \hspace{4 pt} constant \hspace{4 pt} (6.625 \hspace{2 pt} x \hspace{2 pt} 10^{-34} \hspace{2 pt} J \hspace{2 pt} s ) \\ \textbf{c} = speed \hspace{4 pt} of \hspace{4 pt} light \hspace{4 pt} (3.00 \hspace{2 pt} 10^8 \hspace{2 pt} m \hspace{2 pt} s^{-1}) \\ \lambda = wavelength \hspace{4 pt} (m) \\ \textbf{k} = Boltzmann \hspace{4 pt} constant \\ \hspace{10 pt} (1.38 \hspace{4 pt} 10^{-23} \hspace{4 pt} J \hspace{2 pt} K^{-1}) \\ \textbf{T} = temperature \hspace{4 pt} (K) \) |
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$$ \large PV = nRT = NkT $$
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\(\\ \textbf{P} = pressure \hspace{4 pt} (Pa) \\ \textbf{V} = volume \hspace{4 pt} (m^3) \\ \textbf{n} = number \hspace{4 pt} of \hspace{4 pt} moles \hspace{4 pt} of \hspace{4 pt} gas \hspace{4 pt} (mol)\\ \textbf{R} = gas \hspace{4 pt} constant \hspace{4 pt} (8.314 \hspace{2 pt} J \hspace{2 pt} K^{-1} \hspace{4 pt} mol^{-1}) \\ \textbf{T} = temperature \hspace{4 pt} (K) \\ \textbf{N} = number \hspace{4 pt} of \hspace{4 pt} particles\\ \textbf{k} = Boltzmann \hspace{4 pt} constant \\ \hspace{5 pt} (1.38 \hspace{2 pt} x 10^{-23} \hspace{2 pt} m^2 \hspace{2 pt} kg \hspace{2 pt} s^{-2} \hspace{2 pt} K^{-1}) \\\) |
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$$ \large
L = A \sigma T^4 = 4\pi R^2 \sigma T^4 $$
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\(\\ \textbf {L} = luminosity \hspace {4 pt} (W)\\ {\sigma} = Stefan-Boltzmann \hspace {4 pt} constant \hspace {1 pt} (5.67 \hspace {4 pt} x \hspace {4 pt} 10^{-8} \hspace {4 pt} W \hspace {2 pt} m^{-2} \hspace {2 pt} K^{-4}) \\ \textbf{A} = surface \hspace{4 pt} area \hspace {4 pt} (m^2) \\ \textbf {T} = temperature \hspace {4 pt} (K)\\ \textbf {R} = radius \hspace {4 pt} of \hspace {4 pt} the \hspace {4 pt} star \hspace {4 pt} (m)\) |
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$$ \large
F = \frac{L}{A} = \frac{L}{4 \pi R^2} = \sigma T^4 $$
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\(\\ \textbf {F} = flux\hspace {4 pt}(W \hspace{2 pt} m^{-2})\\ \textbf {L} = luminosity \hspace {4 pt} (W)\\ \textbf{A} = surface \hspace{4 pt} area \hspace {4 pt} (m^2) \\ \textbf {R} = radius \hspace {4 pt} of \hspace {4 pt} the \hspace {4 pt} star \hspace {4 pt} (m) \\ {\sigma} = Stefan-Boltzmann \hspace {4 pt} constant \hspace {1 pt} (5.67 \hspace {4 pt} x \hspace {4 pt} 10^{-8} \hspace {4 pt} W \hspace {2 pt} m^{-2} \hspace {2 pt} K^{-4}) \\ \textbf {T} = temperature \hspace {4 pt} (K)\\\) |
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$$ \large m_1 - m_2 = -2.5 \hspace{2 pt} log \left(\frac{F_1}{F_2} \right) $$
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\(\\ {m_\#} = apparent \hspace{4 pt}magnitude \hspace{4 pt} of \hspace{4 pt} star \hspace{4 pt} \# \\ {F_\#} = observed\hspace{4 pt} flux \hspace{4 pt} of \hspace{4 pt} star \hspace{4 pt} \# \) |
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$$ \large F_{elec} = \frac{q_1 q_2}{4 \pi \epsilon_0 \hspace{2 pt} r^2} $$
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\(\\ {F_{elec}} = elecrostatic \hspace{4 pt} force \hspace{4 pt} (N) \\ {q_\#} = charge \hspace{4 pt} of \hspace{4 pt} interacting \hspace{4 pt} particle \hspace{4 pt}\# \hspace{4 pt} (C) \\ {\epsilon_0} = permittivity \hspace{4 pt} of \hspace{4 pt} free \hspace{4 pt} space \hspace{4 pt} \\ \hspace{7 pt} (8.854 \hspace{2 pt} x \hspace{2 pt} 10^{-12} \hspace{2 pt} F \hspace{2 pt} m^{-1} )\\ \textbf{r} = interaction \hspace{4 pt} radius \hspace{4 pt} (m)\) |
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$$ \large U_{coul} = \frac{q_1 q_2}{4 \pi \epsilon_0 \hspace{2 pt} r} $$
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\(\\ {U_{coul}} = Coulomb \hspace{4 pt} barrier \hspace{4 pt} (J) \\ {q_\#} = charge \hspace{4 pt} of \hspace{4 pt} interacting \hspace{4 pt} particle \hspace{4 pt}\# \hspace{4 pt} (C) \\ {\epsilon_0} = permittivity \hspace{4 pt} of \hspace{4 pt} free \hspace{4 pt} space \hspace{4 pt} \\ \hspace{7 pt} (8.854 \hspace{2 pt} x \hspace{2 pt} 10^{-12} \hspace{2 pt} F \hspace{2 pt} m^{-1} )\\ \textbf{r} = interaction \hspace{4 pt} radius \hspace{4 pt} (m)\) |
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