Wheatstone bridge
The arrangement of five capacitors as shown is called Wheat stone bridge.
If 
then points P & Q are at same potential & the bridge is said to be balanced, due to this no charge will flow in the arm PQ & hence arm PQ can be removed & circuit can becomes as shown.
The effective capacitance across the points X & Y for the balanced state of Wheatstone bridge is 
the bridge is not balanced, then the problem can be solved using Kirchoff’s laws.
Parallel grouping of capacitors
1. Potential difference across all the components connected in parallel is same (i.e. V = constant)
2. Charge is among the various capacitors in accordance with q ∝ C i.e. a capacitor of greater capacitance will get more charge & vice versa.
3. Effective capacitance is is given by, 
Conduction
Suppose two charged metal spheres of radii R1 & R2 of different potentials are joined by a metal wire, then charge flows from conductor at higher potential to that at lower potential till both acquire the same potential ‘V’ called common potential. This stage is called steady state & is achieved almost immediately after joining the charged conductors.
1. Common potential at steady state can be calculated using charge conservation i.e. total charge of conductor 1 & 2 before joining
& after joining is same i.e.
q1bj + q2bj = C1 V + C2 V
Or V = V1 aj = V2 aj 
The above relation can be expressed as
2. Charge on each conductor after joining is
As q = C V & C ∝ R, thus bigger sphere gets more charge after conduction.
3. Total energy of the system before joining is
Total energy of the system after joining is
Uaj is found to be smaller than Ubj. The system looses some of its energy in the form of heat, which is given by
Spherical capacitor outer earthed
Suppose a metal sphere of charge +q & radius a is placed concentrically inside a metal shell of radius b. The charge of a induces charge – q on inner surface & +q on its outer surface of shell b. If shell b is earthed then its +q leaks to earth so that the potential of b becomes zero. Due to earthing & induction the final charge distribution will be
Due to this charge distribution electric field will be
1. E = 0 for r < a (i.e. inside a)
2. 
(radially outward) for a ≤ r ≤ b
3. E = 0 for r > b (i.e. outside b)
Due to this charge distribution the potential on a & b will be
On subtracting above relations we get the potential difference between the a & b, it is
Using C = q/V, capacitance for this system becomes
From above result it is clear
1. Let b – a = d is the distance between two spheres & 
, then we can write 
2. in order to have maximum capacitance the radii of two spheres should be as high as possible & separation between them should as small as possible.
3. Suppose the shell b is situated ∞, then we can say sphere a is isolated, then its using b = ∞, we get C = 4 π ε0 a
4. If a & b are very large such that b – a = d & 
, then 
.
5. As 
, thus we can say that the capacitance of a spherical capacitor is always greater than the capacitance of an isolated sphere.
6. If the outer sphere is not earthed & inner sphere is connected to outer by a metallic wire, then entire charge moves outer & capacitance becomes C = 4 π ε0 b.
PPC with a dielectric slab in plates
If a dielectric slab of dielectric constant K, thickness t < d is placed between the plates of a PPC then due to the electric field 
between the capacitor plates the dielectric gets polarized & an electric 
is induced in it, as a consequence net electric field in dielectric is found to be 
.
Due to this field electric field net potential difference across the capacitor plates becomes 
.
Using C = q/V, capacitance of capacitor becomes 
.
1. If the entire space between the capacitor plates is occupied by air or vacuum, then 
. [using K = 1 and t = 0]
2. If the entire space between the capacitor plates is occupied by dielectric, then 
[using t = d]
^Type of physical quantities
Physical quantities can be categorized in following four types:
(a) Scalars (b) Vectors (c) Ratios (d) Tensors
^Dimensionless Variables
Are the physical quantities which have no dimensions but have variable values.
e.g. Angle, Strain, Specific gravity etc.
Capacitor
A capacitor is an arrangement of two conductors (called plates) separated from each other by a dielectric medium & used to trap (or store) electric energy in the form of electric field between its plates.
Point charge
A body of almost no size is called a point body or discrete body i.e. a sphere of radius R → 0 is a point like body. Both field & potential are not defined on a discrete charge, as r → 0 implies E & V → ∞. Outside the point charge field & potential are
Field of a point charge has following properties
1. radially outward if charge is positive & inwards if it is negative.
2. at finite distance around it, field is spherically symmetric & obeys inverse square law i.e. it is not uniform
3. at infinite distance from the it, field is zero
4. maxi. it the charge is placed in vacuum or air.
E & V due to uniformly charged sphere
Charge on a insulated sphere of uniform volume charge density r & radius R is 
,
Charge on a spherical insulated shell or a conducting sphere of uniform surface charge density s & radius R is, Q = σ 4 πR2
Using Gauss law we can write
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