Energy stored in the capacitor
As per work energy theorem, total work done on any body gets converted into energy.
Suppose we apply a force on a pen and displace it to a certain height.
We do some work in raising the pen which is given by the product of force and the displacement.
This work done has actually got stored in the pen in the form of potential energy.
Similarly, if we want to charge a capacitor, we need to do work which gets converted into electrical energy and stored inside the capacitor.
When all the energy is stored inside the capacitor, we say that the capacitor is fully charged.
Let's understand how exactly we can calculate the energy stored in the capacitor.
Consider a capacitor which is initially at time
$t=0$
not charged at all.
Now, we pluck a small fraction of charge
$dQ$
from one plate and deposit it on to the other plate.
This way, a small positive charge develops on one plate while equal negative charge develops on the other plate.
If we go on continuing the plucking and depositing of charge, after some time, let say
$t$
, both plates get oppositely charged.
Let's assume
$Q_{â€²}$
and
$âˆ’Q_{â€²}$
be the charges developed on the two plates after time
$t$
.
Now again, suppose we pluck another small charge
$dQ_{â€²}$
from one plate and deposit it to the other plate.
Let the work done in moving the small charge be
$dW$
which is given by the product of
$V_{â€²}$
and the charge
$dQ_{â€²}$
.
We can further get the work done by replacing the value of
$V_{â€²}$
in terms of
$C$
and
$Q_{â€²}$
, where
$$C$
is capacitance of capacitor.
Let
$Q$
be the total final charge accumulated on the plates when the capacitor is fully charged.
Integrating
$dW$
from the time when the charge on the plates was zero to the time when the final charge on plates is
$Q$
.
And thus we have got the total work done in charging the plates to charge
$Q$
equal to
$2CQ_{2}â€‹$
.
This work done gets stored inside the capacitor in the form of electrical potential energy denoted by
$U$
.
We can write this in other forms also.
Again, we can write this as,
So, finally, we can give all the expressions of the energy stored in the capacitor as,
We can use this stored energy further as per our requirements whenever we need it.
Revision
If we want to charge a capacitor, we need to do work which gets converted into electrical energy and stored inside the capacitor.
The energy stored inside the capacitor is given by,
The End