**Understanding**

➔ Capacitance

➔ Dielectric materials

➔ Capacitors in series and parallel

➔ Resistor–capacitor (RC) series circuits

➔ Time constant

**Applications ****and** **skills**

➔ Describing the effect of different dielectric materials on capacitance

➔ Solving problems involving parallel-plate capacitors

➔ Investigating combinations of capacitors in series or parallel circuits

➔ Determining the energy stored in a charged capacitor

➔ Describing the nature of the exponential discharge of a capacitor

➔ Solving problems involving the discharge of a capacitor through a fixed resistor

➔ Solving problems involving the time constant of an RC circuit for charge, voltage, and current

### Equations

➔ definition of capacitance: *C *=Q/V

➔ combining capacitors in parallel: *C*_{parallel} = *C*1 + *C*2 + …

➔ series: 1/C_{series} = 1/C1 + 1/C2 + 1/C3 …

### 1.3 Capacitance

Capacitor: Two conductors separated from each other by an insulating (dielectric) material (or vacuum).

Storage: Stores electric charge and electric energy.

Design: Formed by two parallel plates with area A and distance d in between.

Capacitance: charge (q) per unit voltage (V) that can be store in a capacitor

Charge distribution: +q on one plate and -q on the other plate

C = Q/V = εA/d

Units: Farad (F)

In a Closed Circuit

**Capacitor:**Electrons move from the plate connected to the positive terminal and transfer to the plate connected to the negative terminal.

Potential difference across capacitor is greater or equal to emf across it.

Combining capacitors: Opposite as with resistors!

In parallel: ∑C = C1 + C2 + C3 +… (same pd across them)

In series: 1/∑C = 1/C1 + 1/C2 + 1/C3 +… (same charge across them)

Energy stored: total work done to charge the capacitor

E = 1/2 CV² = 1/2 QV = 1/2 Q²/C

**The effect of dielectric**

Dielectric material: ε > εo (vacuum), and thus, C > Co

Charge polarization: In the dielectric, there is separation of charges, known as charge polarization.

Small electric field is created, reducing the net electric compared to εo

pd across capacitor is also reduced, since some electric energy is used to align molecules, raising the potential of the negative plate and lowers the potential of the positive plate

**Charging and discharging**

Charging: Accumulating charge on the negative plate

Current starts out large, as if the capacitor was not there, i.e Io = ε/R, but decreases and reaches zero, since the electrons on the negative plate push back new electrons.

When fully charged, no current passes through the capacitor.

Discharging: Capacitor becomes a power source, which is discharge by resistors

Formulas: q = qo e^-t/RC; V = Vo e^-t/RC; I = Io e^-t/RC and Io = qo/RC

Time constant (τ) = RC, is the time scale for discharge (measured in seconds).

Time took for q to decrease to 37% of its original value as it discharges

Similar to half-life in radioactive decay.

### CAPACITORS AND CAPACITANCE

#### ENERGY STORED IN A CAPACITOR

#### SHARING OF CHARGES

### PARALLEL PLATE CAPACITOR

#### EFFECT OF DIELECTRIC ON CAPACITANCE

- The unit farad is quite a big unit for practical purposes. Even the capacitance of a huge body like earth is 711 μF.
- A capacitor is a device which stores charges and produces electricity whenever required.
- If the two plates of a capacitor is connected with a conducting wire, sparking takes place which shows that electrical energy is converted into heat and light energy.
- A capacitor allows A.C. but doesn’t allow D.C. to pass through it.
- The capacitance of a capacitor increases with insertion of a dielectric between its plates and decreases with increase in the separation between the plates.
- The capacitance of a capacitor increases K times if a medium of dielectric constant K is inserted between its plates.
- The energy of a capacitor for a particular separation between the plates is the amount of work done in separating the two plates to that separation if they are made to touch to each other.
- The loss of energy when the two charged conductors are connected by a wire doesn’t depend on the length of the wire.

### SPHERICAL CAPACITOR

- When outer conductor is earthed,

- When inner sphere is earthed,

### CYLINDRICAL CAPACITOR

### COMBINATION OF CAPACITORS

#### SERIES COMBINATION

- In this combination, the positive plate of one capacitor is connected to the negative plate of the other.

- The charges of individual capacitor are equal.
- The potential difference is shared by the capacitors in the inverse ratio of their capacities

- The equivalent capacitance (C) between A and B is

#### PARALLEL COMBINATION

- In this arrangement, +ve plates of all the condensers are connected to one point and negative plates of all the condensers are connected to the other point.
- The Potential difference across the individual capacitor is same.

- The total charge shared by the individual capacitor is in direct ratio of their capacities

- The equivalent capacitance between a and b is ceq = c1 + c2 + c3 + ……..+ cn

- The capacitance of a parallel plate capacitor having a number of slabs of thickness t1, t2, t3 …. and dielectric constant K1, K2, K3 …. respectively between the plates is

- When a number of dielectric slabs of same thickness (d) and different areas of cross-section A1, A2, A3 … having dielectric constants K1, K2, K3, …. respectively are placed between the plates of a parallel plate capacitor then the capacitance is given by

- When five capacitors are connected in wheatstone bridge arrangement as shown, such that, the bridge is balanced and C5 becomes ineffective. No charge is stored on C5. Therefore C1, C2 and C3, C4 are in series. The two series combinations are in parallel between A and C. Hence equivalent capacitance can be calculated.

### RELATION BETWEEN THREE ELECTRIC VECTORS

### EFFECT OF FILLING DIELECTRIC WITH BATTERY CONNECTED

### EFFECT OF FILLING A DIELECTRIC IN A CAPACITOR AFTER DISCONNECTION OF BATTERY

### CHARGING AND DISCHARGING A CAPACITOR

#### CHARGING A CAPACITOR

#### DURING THE PERIOD OF CHARGING

- The charge on the capacitor increases from ‘zero’ to the final steady charge.
- The potential difference developed across the capacitor opposes the constant potential difference of the source.
- The charge on the capacitor ‘grows’ only as long as the potential difference of source is greater than the potential difference across the capacitor. This transport of the charge from the source to the capacitor constitutes a transient current in the circuit.
- As the charge on the capacitor increases, more energy is stored in the capacitor.
- When the capacitor is fully charged, potential difference across the capacitor is equal to the potential difference of the source and the transient current tends to zero.

i.e.

- At t = 0, q = 0.
- When t increases, q increases.
- As
- At t = CR [‘CR’ has dimensions of time]

#### DISCHARGING OF A CAPACITOR

- the initial condition, q = Q0 at t = 0 and
- the final condition, q = 0 at ,

- If n small drops each having a charge q, capacity ‘C’ and potential V coalesce to form a big drop, then
- the charge on the big drop = nq
- capacity of big drop = n1/3 C
- potential of big drop = n2/3 V
- potential energy of big drop = n5/3 U
- surface density of charge on the big drop = n1/3 × surface density of charge on one small drop.

- Charged soap bubble : Four types of pressure act on a charged soap bubble.
- Pressure due to air outside the bubble PO, acting inwards.
- Pressure due to surface tension of soap solution PT, acting inwards.
- Pressure due to air inside the bubble, Pi, acting outwards.
- Electric pressure due to charging, Pe =, acting outwards.

- Force of attraction between the plates of a parallel plate capacitor =

- Uses of capacitor :
- In LC oscillators
- As filter circuits
- Tuner circuit in radio etc.

- The total energy stored in an array of capacitors (in series or in parallel) is the sum of the individual energies stored in each capacitor.

### COMBINATION OF CAPACITOR : EQUIVALENT CAPACITANCE

### SOME METHODS OF FINDING EQUIVALENT CAPACITANCE