Description→
When a capacitor is charged, the work is done by charged battery (i.e the chemical energy of the battery is used to charge the capacitor). As the capacitor gets charged, the potential difference between its plates increases. Due to this increase in the potential difference between the plates, the battery has to give the same amount of charge to the capacitor. Because of that, the battery has to do more and more work.
The total amount of work done in charging the capacitor is stored in the form of electric potential energy in between the capacitor plates. This energy is retrieved as heat when the capacitor is discharged through a resistance.
Derivation→
Let us consider, a capacitor of capacitance $C$with a potential difference of $V$ between the plates.
In the process of charging, electrons are transferred from the positive to negative, unit each plate acquires an amount of charge $q$. Suppose during the process of charging, we increase the charge from $q'$ to $q'+dq'$ by transferring an amount of negative charge $dq'$ from the positive to the negative plate. the work done
$dw=V' \: dq'$
$dw=\frac{q'}{C} \: dq'$
Therefore, the total work done in charging the capacitor from the uncharged state (i.e. zero charge in the capacitor) to the final charge $q$ will be
$W=\int_{0}^{W} dw$
$W=\int_{0}^{q} \frac{q'}{C} \: dq'$
$W=\frac{1}{2}\left[ \frac{q'^{2}}{C} \right]^{q}_{0} $
$W=\frac{1}{2}\left[ \frac{q^{2}}{C} \right]$
$W=\frac{1}{2}\:C \: V^{2}$
This work done is stored in the form of potential energy $U$ within the capacitor. Thus
$U=\frac{1}{2}\:C \: V^{2} $
$U=\frac{1}{2}\frac{q^{2}}{C}$
Energy Stored in Combination of Capacitor→
If many capacitors are combined in series, or in parallel, the total potential energy stored in either combination is equal to the sum of the potential energies stored in the individual capacitor. This follows from the combination of the capacitor's expressions:
The potential energy is stored in a series combination→
The potential energy stored in a series combination (here $q$ is constant) is
$U=\frac{1}{2}\frac{q^{2}}{C}$
For series combination the capacitance of capacitor i.e. $\frac{1}{C}=\frac{1}{C_{1}}+\frac{1}{C_{2}}+\frac{1}{C_{2}}+..........$. Now substitute the value of $\frac{1}{C}$ in above equation
$U=\frac{1}{2}q^{2} \frac{1}{C_{1}}+\frac{1}{C_{2}}+\frac{1}{C_{2}}+.......... $
$U= \left( \frac{1}{2}q^{2} \frac{1}{C_{1}}+\frac{1}{2}q^{2} \frac{1}{C_{2}}+ \frac{1}{2}q^{2} \frac{1}{C_{2}}+.......... \right)$
$U=U_{1}+U_{2}+U_{3}+.........$
The potential energy is stored in a parallel combination→
The potential energy is stored in a parallel combination (here $V$ is constant) is
$U=\frac{1}{2}CV^{2}$
For parallel combination the capacitance of capacitor i.e. $C=C_{1}+C_{2}+C_{3}+......$. Now substitute the value of $C$ in the above equation
$U=\frac{1}{2} \left( C_{1}+C_{2}+C_{3}+........... \right) V^{2}$
$U= \frac{1}{2} C_{1} V^{2} + \frac{1}{2} C_{2} V^{2} +\frac{1}{2} C_{3}V^{2} +..... $
$U=U_{1}+U_{2}+U_{3}+......$
Angle of Acceptance → If incident angle of light on the core for which the incident angle on the core-cladding interface equals the critical angle then incident angle of light on the core is called the "Angle of Acceptance. Transmission of light when the incident angle is equal to the acceptance angle If the incident angle is greater than the acceptance angle i.e. $\theta_{i}>\theta_{0}$ then the angle of incidence on the core-cladding interface will be less than the critical angle due to which part of the incident light is transmitted into cladding as shown in the figure below Transmission of light when the incident angle is greater than the acceptance angle If the incident angle is less than the acceptance angle i.e. $\theta_{i}<\theta_{0}$ then the angle of incidence on the core-cladding interface will be greater than the critical angle for which total internal reflection takes place inside the core. As shown in the figure below Transmission of lig
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