IEMDaily - Video Lecture Notes

Definition

Spherical capacitor with dielectrics

C = \frac{4 π ϵ_{0} K_{1} K_{2} a b c}{K_{1} a (c - b) + K_{2} c (b - a)}

Formula

Capacitors in series

The effective capacitance of capacitors in series is
The charge q flowing through the caapcitor is same in series combination, only voltage is divided.

V_{1} = \frac{q}{C_{1}}, V_{2} = \frac{q}{C_{2}}, V_{3} = \frac{q}{C_{3}}

V = V_{1} + V_{2} + V_{3} + . . . . .

C_{e q} = \frac{q}{V} = \frac{1}{\frac{1}{C_{1}} \frac{1}{C_{2}} \frac{1}{C_{3}}}

{\frac{1}{C}}_{e q} = {\frac{1}{C}}_{1} + {\frac{1}{C}}_{2} + . . . . + {\frac{1}{C}}_{n}

Formula

Capacitors in parallel

The effective capacitance of capacitors in parallel is :
In parallel combination voltage remains same across caapcitors only charge gets divided. so,

q_{1} = C_{1} V, q_{2} = C_{2} V, q_{3} = C_{3} V . . . . .

q = q_{1} + q_{2} + q_{3} + . . . . .

= (C_{1} + C_{2} + C_{3}) V

C_{e q} = C_{1} + C_{2} + . . . . . + C_{n}

Example

Capacitors in series and parallel

Three capacitors each of

6 μ F

are connected together in series and then connected in series with the parallel combination of three capacitors of

2 μ F, 4 μ F

and

2 μ F

. The total combined capacity is found as follows:

Effective capacitor of

6 μ F

in series

\frac{1}{C_{e f f}} = \frac{1}{6} + \frac{1}{6} + \frac{1}{6}

C_{e f f} = 2 μ F

Effective capacitor of parallel capacitor

C_{e f f} = C_{1} + C_{2} + C_{3}

C_{e f f} = 2 + 4 + 2

C_{e f f} = 8 μ F

\frac{1}{C_{e f f}} = \frac{1}{2 μ F} + \frac{1}{8 μ F}

\Rightarrow C_{e f f} = 1.6 μ F

Example

Energy stored in a combination of capacitors

A 4

μ F

capacitor charged to

50 V

is connected to another 2

μ F

capacitor charged to

100 V

. The total energy of combination is found as follows:
After connecting capacitors, the potential of two capacitor will be same:
The charge will be conserved

Q_{f} = Q_{i}

C_{1} V_{1} + C_{2} V_{2} = (C_{1} + C_{2}) V

4 \times 50 + 2 \times 100 = (4 + 2) V

V = \frac{400}{6} = \frac{200}{3}

\Rightarrow

Final energy

= \frac{1}{2} (C_{1} + C_{2}) V^{2}

Total energy

= 13.33 \times 10^{- 3} J

Definition

Charging and Discharging of capacitors

Charging:When a capacitor is connected to a battery, positive charge appears on one plate and negative charge on the other. The potential difference between the plates ultimately becomes equal to the emf of the battery. The whole process takes some time and during this time there is an electric current through the connecting wires and the battery.
$q = ϵ C (1 - e^{\frac{- t}{C R}})$ where

q

is the charge on the capacitor at time

t, C R

is called the time constant,

ϵ

is the emf of the battery.

Discharging:
If the plates of a charged capacitor are connected through a conducting wire, the capacitor gets discharged. Again there is a flow of charge through the wires and hence there is a current.

q = Q e^{\frac{- t}{C R}}

where

q

is the charge on the capacitor at time

t, Q

is the charge on the capacitor at time

t = 0

and

C R

is called the time constant,

Formula

Energy stored in a capacitor

The energy stored in a capacitor is

U = \frac{1}{2} \frac{Q^{2}}{C} = \frac{1}{2} Q V = \frac{1}{2} C V^{2}

Example

Solving capacitor circuits

Example:
In the given figure, find charge across each capacitor in steady state.
Solution:
Replace capacitors of

3 μ F

and

6 μ F

by a single capacitor of

9 μ F

as equivalent capacitance of capacitors in parallel is the sum of each capacitance.
Since capacitors are in series, charge on each capacitor is same by law of conservation of charge.