- #1
Reshma
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The Charge density of an electron cloud for a Hydrogen atom is given by:
[tex]\rho (r) = \left(\frac{q}{\pi a^3}\right)e^{\frac{-2r}{a}}[/tex]
Find its polarizability([itex]\alpha [/itex]).
My work:
Dipole moment p is:
[tex]\vec p = \alpha \vec E[/tex]
I need to calculate the electric field first.
The electric field is given by Gauss's law:
[tex]\vec E = \left(\frac{1}{4\pi \epsilon_0}\right)\frac{Q_{total}}{r^2}\hat r[/tex]
[tex]Q_{total} = \int_{0}^{r} \rho (r)d\tau[/tex]
[tex]Q_{total} = \int_{0}^{r} \left(\frac{q}{\pi a^3}\right) e^{\frac{-2r}{a}} 4\pi r^2 dr[/tex]
[tex]Q_{total} = \frac{4q}{a^3} \int^{r}_{0} e^{\frac{-2r}{a}} r^2 dr[/tex]
How is this integral evaluated?
[tex]\rho (r) = \left(\frac{q}{\pi a^3}\right)e^{\frac{-2r}{a}}[/tex]
Find its polarizability([itex]\alpha [/itex]).
My work:
Dipole moment p is:
[tex]\vec p = \alpha \vec E[/tex]
I need to calculate the electric field first.
The electric field is given by Gauss's law:
[tex]\vec E = \left(\frac{1}{4\pi \epsilon_0}\right)\frac{Q_{total}}{r^2}\hat r[/tex]
[tex]Q_{total} = \int_{0}^{r} \rho (r)d\tau[/tex]
[tex]Q_{total} = \int_{0}^{r} \left(\frac{q}{\pi a^3}\right) e^{\frac{-2r}{a}} 4\pi r^2 dr[/tex]
[tex]Q_{total} = \frac{4q}{a^3} \int^{r}_{0} e^{\frac{-2r}{a}} r^2 dr[/tex]
How is this integral evaluated?
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