Obtain the expression for the energy stored per unit volume in a charged capacitor.
Obtain the expression for the energy stored per unit volume in a charged capacitor.
Energy stored per unit volume is known as energy density. The energy stored in capacitor,
$\mathrm{U}=\frac{1}{2} \frac{\mathrm{Q}^{2}}{\mathrm{C}}$
$=\frac{1}{2} \frac{(\sigma \mathrm{A})^{2}}{1} \times \frac{d}{\epsilon_{0} \mathrm{~A}} \text { where } \mathrm{Q}=\sigma \mathrm{A} \text { and } \mathrm{C}=\frac{\epsilon_{0} \mathrm{~A}}{d}$
$=\frac{\sigma^{2} \mathrm{~A} d}{\epsilon_{0}}$
$\text { but } \frac{\sigma}{\epsilon_{0}}=\mathrm{E}$
$\mathrm{U}=\frac{1}{2} \mathrm{E}^{2} \epsilon_{0} \times \mathrm{A} d$
But $\mathrm{A} d$ is the volume of the region between the plates.
$\therefore \frac{\mathrm{U}}{\mathrm{Ad}}=\frac{1}{2} \epsilon_{0} \mathrm{E}^{2}$ is a energy per unit volume.
It is denoted by $\rho_{\mathrm{E}}$ or ' $u$ '. $\therefore$ Energy per unit volume, $\rho_{\mathrm{E}}=\frac{1}{2} \epsilon_{0} \mathrm{E}^{2}$
Similar Questions
The energy density $u$ is plotted against the distance $r$ from the centre of a spherical charge distribution on a $log$-$log$ scale. The slope of obtianed straight line is :
The energy density $u$ is plotted against the distance $r$ from the centre of a spherical charge distribution on a $log$-$log$ scale. The slope of obtianed straight line is :
A parallel plate capacitor has a uniform electric field $E$ in the space between the plates. If the distance between the plates is $d$ and area of each plate is $A,$ the energy stored in the capacitor is
A parallel plate capacitor has a uniform electric field $E$ in the space between the plates. If the distance between the plates is $d$ and area of each plate is $A,$ the energy stored in the capacitor is
- [AIPMT 2011]
A $20\,F$ capacitor is charged to $5\,V$ and isolated. It is then connected in parallel with an uncharged $30\,F$ capacitor. The decrease in the energy of the system will be.......$J$
A $20\,F$ capacitor is charged to $5\,V$ and isolated. It is then connected in parallel with an uncharged $30\,F$ capacitor. The decrease in the energy of the system will be.......$J$
Two small spheres each carrying a charge $q$ are placed $r$ metre apart. If one of the spheres is taken around the other one in a circular path of radius $r$, the work done will be equal to
Two small spheres each carrying a charge $q$ are placed $r$ metre apart. If one of the spheres is taken around the other one in a circular path of radius $r$, the work done will be equal to
A capacitor of capacitance $6\,\mu \,F$ is charged upto $100$ $volt$. The energy stored in the capacitor is........$Joule$
A capacitor of capacitance $6\,\mu \,F$ is charged upto $100$ $volt$. The energy stored in the capacitor is........$Joule$