CAPACITOR(Notes)
1. CAPACITANCE OF AN ISOLATED SPHERICAL CONDUCTOR :
C = 4 0r
R in a medium C = 4 0 R in air
* This sphere is at infinite distance from all the conductors .
* The capacitance C = 4 0 R exists between the surface of the sphere & earth .
2. SPHERICAL CAPACITOR :
It consists of two concentric spherical shells as shown in figure. Here capacitance of region
between the two shells is C1
and that outside the shell is C2
. We have
Depending on connection, it may have different combinations of C1
and C2
.
3. PARALLEL PLATE CAPACITOR :
(i) UNIFORM DI-ELECTRIC MEDIUM :
If two parallel plates each of area A & separated by a distance d are charged with
equal & opposite charge Q, then the system is called a parallel plate capacitor & its capacitance is
given by,
C =
in a medium ; C = 0 A
d
with air as medium
This result is only valid when the electric field between plates of capacitor is constant.
(ii) MEDIUM PARTLY AIR : C =
When a di-electric slab of thickness t & relative permittivity r
is
introduced between the plates of an air capacitor, then the distance between
the plates is effectively reduced by t
irrespective of the position of
the di-electric slab .
(iii) COMPOSITE MEDIUM : C =
4. CYLINDRICAL CAPACITOR :
It consist of two co-axial cylinders of radii a & b, the outer conductor is earthed .
The di-electric constant of the medium filled in the space between the cylinder is
r
. The capacitance per unit length is C =
5. CONCEPT OF VARIATION OF PARAMETERS:
As capacitance of a parallel plate capacitor isC =
, if either of k, A or d varies in the region between
the plates, we choose a small dc in between the plates and for total capacitance of system.
If all dC's are in series
, If all dC's are in parallelCT
= dC
6. COMBINATION OF CAPACITORS :
(i) CAPACITORS IN SERIES :
In this arrangement all the capacitors when uncharged get the same charge
Q but the potential difference across each will differ (if the capacitance are
unequal).
Ceq.
1
= C1
1
+
C2
1
+ C3
1
+ ........ + Cn
1
.
(ii) CAPACITORS IN PARALLEL :
When one plate of each capacitor is connected to the positive
terminal of the battery & the other plate of each capacitor is
connected to the negative terminals of the battery, then the
capacitors are said to be in parallel connection.
The capacitors have the same potential difference, V but the
charge on each one is different (if the capacitors are unequal).
C
eq.
= C1 + C2 + C3 + ...... + Cn
.
7. ENERGY STORED IN A CHARGED CAPACITOR :
Capacitance C, charge Q & potential difference V; then energy stored is
U = 1/2(CV^2)
. This energy is stored in the electrostatic field set up in the di-electric
medium between the conducting plates of the capacitor .
8. HEAT PRODUCED IN SWITCHING IN CAPACITIVE CIRCUIT
Due to charge flow always some amount of heat is produced when a switch is closed in a circuit which
can be obtained by energy conservation as –
Heat = Work done by battery – Energy absorbed by capacitor.
9. SHARING OF CHARGES :
When two charged conductors of capacitance C1
& C2
at potential V1
& V2
respectively are
connected by a conducting wire, the charge flows from higher potential conductor to lower potential
conductor, until the potential of the two condensers becomes equal. The common potential (V)
after sharing of charges;
V =
net ch e
net capacitance
.
charges after sharing q1
= C1V & q2
= C2V. In this process energy is lost in the connecting wire
as heat . This loss of energy is Uinitial Ureal =
.
10. REMEMBER :
(i) The energy of a charged conductor resides outside the conductor in its EF, where as in a condenser
it is stored within the condenser in its EF.
(ii) The energy of an uncharged condenser = 0 .
(iii) The capacitance of a capacitor depends only on its size & geometry & the di-electric between the
conducting surface . (i.e. independent of the conductor, like, whether it is copper, silver, gold etc)
