Capacitance

1. Capacitor:

“Capacitors are components of electrical circuits that temporarily store electric charge”.

• The addition of a capacitor into a circuit has two possible effects: either introducing a time delay into the circuit; or storing electrical energy for a short period of time.
• Capacitors are used extensively in electrical and electronic timing circuits, in power circuits, for smoothing electrical signals and as part of the signal-receiving circuits found in radios.
• Modern capacitors consist of two parallel conducting plates (usually made of metal foils, films or coatings) separated by a thin insulating layer known as a dielectric (generally made from thin plastic films, electrolytes, ceramics or metal oxides).
• 
  Figure 1 Different types of capacitors
• There are several different circuit symbols for capacitors depending on their type, although they are all based on the same simple pattern shown in Figure 1.
• 
  Figure 2 Some types of capacitor’s symbols
• A potential difference from a battery or a power supply connected across the metal plates causes electrons to flow off one plate, back through the battery and onto the second plate (Figure 3).
• 
  Figure 3 A power supply connected across the metal plates
• One plate becomes positively charged (where electrons are moved), while the plate with the excess of electrons becomes negatively charged. If the capacitor is then disconnected from the source of potential difference, the charge will stay on the plates until a conducting pathway allows the excess electrons to flow off the negatively charged plate and back onto the positive plate, until the two plates have equal charge again (Figure 4).
• 
  Figure 4 Flow of electrons in circuit
• The conducting pathway could be a different part of the circuit (controlled by a switch) or the charge could gradually leak away to the surroundings.

• ⇒ Capacitance:

• The ability of any object to store charge is called capacitance.
• Capacitance is given the symbol C, and the S1 unit is the farad (F).
• The capacitance of a capacitor depends on the area of the metal plates, the distance between the plates and the electrical properties of the material separating the plates.
• The amount of charge, Q, that can be stored on a capacitor depends on the size of the capacitance, C, and the potential difference, V, across the capacitor causing the separation of the charge:
• [math] Q \propto V \\ Q = CV [/math]

• The capacitance of a capacitor can then be defined by

• [math] C = \frac{Q}{V} [/math]
• So, one farad is equal to one coulomb per volt ( [math] C.V^{-1} [/math]). Actually, 1 F is quite a large capacitance, and useful ‘real-life’ capacitors have capacitances measured in microfarads (µF), nanofarads (nF) or picofarads (pF).

2. Parallel plate-capacitor:

• The capacitance of a parallel-plate capacitor depends on the area of the plates, their distance apart and the ability of the insulating material between the plates to separate the charge, a property known as permittivity.
• The permittivity of a material is the resistance of the material to an electric field passing through it.
• If the permittivity is high, then a larger charge can be stored on the plates for any given potential difference across them.
• The permittivity of capacitor insulating materials is always measured relative to the permittivity of free space (vacuum),[math] \varepsilon_0 [/math] , using a relative permittivity ([math] \varepsilon_r [/math]), sometimes called the dielectric constant of the material.
• The total absolute permittivity of an insulator is therefore given by the product [math] \varepsilon_0 \varepsilon_r [/math].
• The permittivity of free space [math] =8.854 * 10^{-12} F.m^{-1} [/math].
• Table 1 gives the relative permittivity of a selection of materials commonly used in the construction of capacitors.
• Table1 Relative permittivity of some materials

• Material	Relative permittivity (dielectric constant)
  Ceramic (ZnMg)TiO2	32
  Polyester	2.8-4.5
  Polystyrene	2.5-2.7
  Aluminium oxide (electrolyte)	9.8

• Figure 5 Charges flow into dielectric ceramic material when circuit close
• The capacitance of a parallel-plate capacitor is given by
• [math] C = \frac{\varepsilon_0 \varepsilon_r A}{d} [/math]
• Where A is the area of the plates and d is their separation. Since the capacitance is also given by
• [math] C = \frac {Q}{V} [/math]
• It follows that
• [math] \frac{Q}{V} = \frac{\varepsilon_0 \varepsilon_r A}{d} \\
  \frac{Q}{A} = \frac{\varepsilon_0 \varepsilon_r V}{d} = \frac{\varepsilon_0 \varepsilon_r V}{d}[/math]

• So, the charge density on each plate is proportional to  [math] \frac{V}{d} [/math] which is the electric field E.

3. The action of a simple polar molecule:

• A simple polar molecule is a molecule that has a permanent electric dipole moment, meaning it has a slightly positive charge on one end and a slightly negative charge on the other. This is due to the unequal sharing of electrons between atoms in the molecule.
• Figure 6 polar charge molecules
• Examples of simple polar molecules include:
  1. Water (H2O)
  2. Carbon monoxide (CO)
  3. Ammonia (NH3)
  4. Sulfur dioxide (SO2)
  5. Hydrogen chloride (HCl)
• Characteristics of simple polar molecules:
  1. Asymmetrical shape
  2. Unequal electron sharing between atoms
  3. Permanent electric dipole moment
  4. Slightly positive charge on one end (δ+)
  5. Slightly negative charge on the other end (δ-)
• Some molecules, such as water, are called polar molecules because the opposite ends of the molecule have opposite charges (Figure 7).
• 
  Figure 7 Vibrating water polar molecules
• When water molecules form, the hydrogen atoms become slightly positively charged, and the oxygen atom becomes slightly negatively charged. (There is a covalent bond between the hydrogen and oxygen atoms in a water molecule, but the shared electrons in the bond are attracted more towards the oxygen atom than they are towards the hydrogen atoms.)
• In a polar molecule, the overall charge of the molecule is zero, but different ‘ends’ of the molecule may have opposite charges.
• Because one end of a polar molecule causes the metal to have an opposing charge, the molecule is drawn to the metal surface and sticks to it with ease.

• ⇒ Dielectric material:

• A dielectric material is a type of material that is a poor conductor of electricity, but can store electric charge.
• Dielectrics are insulators that can be polarized, meaning they can exhibit a separation of electric charge within the material.
• Most of the dielectric materials used to construct capacitors are solids, and the atoms and molecules are fixed within the structure.
• The electric field produced between the two plates of a capacitor will cause the charged particles in a liquid or a gas to align themselves in the direction of the field (Figure 8 (a)).
• The separated charge in a polar molecule is particularly able to align itself with the field between two capacitor plates.
• If the electric field between the plates is suddenly reversed, the polar molecule will rotate and align itself with the direction of the electric field again (Figure 8 (b)). Alternating the electric field between the two plates will cause a polar molecule, such as water, to continuously rotate between them (called dipole rotation).

Figure 8 (b) Polar molecules rotating in an alternating microwave field

• This increases its kinetic energy, and causes it to collide with other adjacent molecules and atoms.
• These then acquire more kinetic energy and move in random directions, increasing their temperature and so dissipating the energy as heat.
• The alternating field between the plates can be produced by a microwave emitter, such as the magnetron inside a microwave oven.
• The frequency of the microwaves is tuned so that it rotates water molecules within food – causing the food to heat up rapidly.

4. Interpretation of the area under a graph of charge against pd.

• When a capacitor is charged up, the pd. from the electricity supply (or the energy per unit charge) causes electrons to flow off one plate, through the external circuit and onto the other plate.
• This separation of charge is kept steady provided that the pd. is continuously applied, and that there is no leakage of charge.
• Once the pd. is removed, and a complete discharging circuit is connected to the capacitor, the electrical energy stored by the separated charge can be released as the electrons flow back off the negatively charged plate and back onto the positively charged plate.
• 
  Figure 9 (a) Graph of Q versus V for a capacitor (b) The energy stored on a capacitor is equivalent to the area under a Q against V graph.
• If the pd applied to the plates is increased, more charge and therefore energy is stored on the plates. A graph of potential difference against charge for a capacitor is shown in Figure 9 (a).

5. [math] E = \frac{1}{2} QV = \frac{1}{2} CV^2 = \frac{1}{2} \frac{Q^2}{C} [/math]

• You will remember from the definition of potential difference that
• [math] V = \frac{W}{Q} [/math]
• where W is the amount of work done per unit charge, Q.
• In the context of a capacitor, the potential difference V is the amount of work done in moving unit charge off one plate and onto another plate.
• At any potential difference V, the work done moving an amount of charge is therefore
• [math] W = V \Delta Q [/math]
• This is represented by the shaded area in the graph in Figure 9 (a).
• The total energy E stored on the capacitor, charging the capacitor from empty up to a charge of Q at a potential difference V, is calculated by adding up all the similar shaped areas from Q = 0 up to a charge Q.
• In other words, this is the whole area under the graph up to Q. Because the shape is a triangle (Figure 9 (b)),
• [math] E = \frac{1}{2} QV [/math]
• But
• [math] Q = CV [/math]
• So 
• [math]  E = \frac{1}{2} (CV)V [/math]
• [math] E = \frac{1}{2} CV^2 \qquad (1)[/math]
• [math] Q = CV \\ V = \frac{Q}{C} [/math]

• Put in equation 1

• [math] E = \frac{1}{2} C \left(\frac{Q}{C}\right)^2 [/math]
• and
• [math] E = \frac{1}{2} \frac{Q^2}{C} [/math]