Volume of Truncated Cone (Frustrum) and Regular Cone

.

Volume of Truncated Cone

.

The small elemental volume dV has a height of dh and radius r, \[\frac{R_1}{H+H_2}=\frac{R_2}{H_2}=\frac{r}{h+H_2}\] \[R_1H_2=R_2H+R_2H_2\] \[(R_1-R_2)H_2=R_2H\] \[H_2=\frac{R_2H}{R_1-R_2}\] To write changing radius r as a function of height h, \[\frac{R_2}{H_2}=\frac{r}{h+H_2}\] \[\frac{R_1-R_2}{H}=\frac{r-R_2}{\,h+\dfrac{R_2H}{R_1-R_2}\,}\] \[r=\frac{h(R_1-R_2)+R_2H}{H}\] \[r=R_2+\frac{(R_1-R_2)}{H}\,h\] Differential volume element Cross-sectional area: \[A(h)=\pi \,r^2\] Differential volume: \[dV=A(h)\,dh\] \[dV=\pi r^2\,dh\] Substituting for r: \[dV=\pi\left[R_2+\frac{(R_1-R_2)}{H}h\right]^2dh\] Expanding: \[dV=\pi\left[R_2^2+\frac{(R_1-R_2)^2}{H^2}h^2+\frac{2R_2(R_1-R_2)}{H}h\right]dh\] Total volume of the frustum \[V=\int_0^H\pi\left[R_2^2+\frac{(R_1-R_2)^2}{H^2}h^2+\frac{2R_2(R_1-R_2)}{H}h\right]dh\] Integrating: \[V=\pi\left[R_2^2h+\frac{(R_1-R_2)^2}{H^2}\frac{h^3}{3}+\frac{2R_2(R_1-R_2)}{H}\frac{h^2}{2}\right]_{0}^{H}\] \[V=\pi H\left[R_2^2+\frac{(R_1-R_2)^2}{3}+R_2(R_1-R_2)\right]\] Combining terms: \[V=\frac{\pi H}{3}\left[3R_2^2+(R_1-R_2)^2+3R_2(R_1-R_2)\right]\] Expanding: \[V=\frac{\pi H}{3}\left[R_1^2+R_1R_2+R_2^2\right]\] Final Result

\[\boxed{V=\frac{\pi H}{3}\left(R_1^2+R_1R_2+R_2^2\right)}\]

Solid Sphere

.

Uniform Volume Charge density, so \[\rho = \frac{Q}{\frac{4}{3}\pi R^3}=\frac{q}{\frac{4}{3}\pi r^3}=\frac{dq}{4\pi r^2 dr}\] Small amount of work done or energy dU to bring the small elemental charge dq to the sphere of radius r and charge q is, \[dU=k \frac{q \, dq}{r}\] \[dU=k \left(\frac{Q\,r^3}{R^3}\right) \left(\frac{3Q \, r^2 \, dr}{R^3}\right) \frac{1}{r}\] \[U=k \frac{3Q^2}{R^6}\int_0^R r^4dr\] \[U=k\frac{3Q^2}{R^6}\left[\frac{r^5}{5}\right]_0^R\]

\[\boxed{U=k \frac{3}{5}\frac{Q^2}{R}}\]

Self Energy of two Charged Solid Spheres

Self Energy of Two Concentric Charged Shells

.

\[\boxed{U=k \frac{Q_1^2}{2R_1} + k \frac{Q_2^2}{2R_2} + k\frac{Q_1 Q_2}{R_2} } \]

Self Energy of Two Charged Solid Spheres

.

\[\boxed{ U = k \frac{3}{5}\frac{Q_1^2}{R_1} + k \frac{3}{5}\frac{Q_2^2}{R_2} + k\frac{Q_1 Q_2}{r} }\]