1 Million Means: 1 million in numerical is represented as 10,00,000. The Indian equivalent of a million is ten lakh rupees. It is not a...

1 Million Means: 1 Million in Rupees, Lakhs and Crores

June 5, 2024**Capacitor**: Capacitors are the device that causes the movement of a ceiling fan. It is owing to the capacitor that a ceiling fan spins. It is the magnetic torque generated by the capacitor that causes it to spin.

We use capacitors in various daily use appliances like alarm clocks, phone screens, stereo systems, flash cameras etc. In addition, we need capacitors to make sensors, rectifiers, amplifiers or LC oscillators to perform tasks like tuning, filtering, and timing.

A **capacitor** is a passive device with two terminals, capable of storing electrical energy in an electric field, much like a small rechargeable battery. It usually has two metal plates on which electrical charges of opposite nature are induced. This generates an electric field in between the plates and develops a potential difference across the capacitor.

A system of two conductors separated by an insulator is known as a capacitor. The insulator can be air or any dielectric medium like mica, ceramic or plastic film. The capacitor passes the alternating current and blocks the direct current. It is an electronic component that stores electricity and releases it into a circuit when required. An ideal capacitor does not dissipate energy, although real-life capacitors do dissipate a small amount of energy.

- Two parallel lines separated by a small distance represent a fixed capacitor. Therefore, it has a fixed value of capacitance.
- One curved, separated by a small distance from a straight plate, represents a polarized capacitor.
- Two parallel plates separated by a small distance with an arrow across the plates represent a variable capacitor. Therefore, it has a variable value of capacitance.

When we connect a battery across a capacitor, an electric field develops across the dielectric, causing a net positive charge to collect on one conductor and an equal net negative charge to collect on the other conductor. As a result, no current flows through the dielectric, and the total charge of the capacitor is zero.

The electric field generated in the region and the potential difference (which is the work done per unit positive charge in taking a small test charge from one conductor to the other) is proportional to the charge developed on the conductor. If \(Q\) represents the magnitude of the charge induced on the conductor and \(V\) be the potential applied across the conductors, then for the given capacitor, the ratio \(\frac {Q}{V}\) is a constant:

\(C = \frac{Q}{V}\)

The constant \(C\) is called the capacitance of a capacitor.

It is the ability of a component or circuit to receive, collect, and store energy in the form of an electrical charge. It is equal to the ratio of the amount of electric charge stored on a conductor to a difference in electric potential for a capacitor. The capacitance only depends on the geometrical shape of the capacitor or the nature of the dielectric medium.

SI unit of charge is coulomb \(C,\) and the SI unit of voltage is volt \(V.\) SI unit of capacitance is, therefore, Coulomb per Volt, generally known as farad, named for English physicist Michael Faraday and we use \(F\) to represent it. Thus, one farad is equal to the capacitance of a capacitor with one Coulomb charge and one Volt potential difference across it.

One farad is a substantial amount of capacitance, and we generally need smaller units to measure capacitance in our day-to-day lives. Some of the smaller units of capacitance are:

1. Millifarad: \(1\,mF = {10^{ – 2}}\,F\)

2. Microfarad: \(1\,\mu F = {10^{ – 6}}\,F\)

3. Nanofarad: \(1\,nF = {10^{ – 9}}\,F\)

4. Picofarad: \(1\,pF = {10^{ – 12}}\,F\)

A dielectric is an insulating material or a poor conductor of electric current. No current flows through a dielectric when it is placed in an electric field**. **When a dielectric is placed inside a capacitor, it decreases the electric field within the capacitor, reduces the voltage and increases the capacitance—a capacitor with dielectric stores the same charge as one without a dielectric at a comparatively lower voltage. Therefore, a capacitor with a dielectric in it is more effective.

When a dielectric with dielectric constant \(K\) is placed between the plates of a parallel plate capacitor, its capacitance becomes :

\(C = \frac{{{\varepsilon _0}KA}}{d}\)

1.For vacuum, \(K = 1\)

2.In any other medium of permittivity \(\varepsilon ,\,K = \frac{\varepsilon }{{{\varepsilon _0}}}\)

3.Thus, \(K = \frac{c}{{{c_0}}}\) and \(K > 1\)

1. **Parallel Plate Capacitor: **It is the simplest form of capacitor. It contains two significant plane parallel conducting plates separated by a small distance.

The capacitance \(C\) of a parallel plate capacitor is

\(C = \frac{{{\varepsilon _0}A}}{d}\)

Where \(A\) is the area of the plate, \(d\) is the distance between the plates and \({\varepsilon _0}\) represents the permittivity of space.

2. **Spherical Capacitor: **A spherical capacitor contains a solid or hollow spherical conductor of radius \(a,\) surrounded by another hollow concentric spherical conductor of radius \(b,\) such that \(a < b.\)

The capacitance \(C\) of a spherical capacitor is

\(C = \frac{{4\pi {\varepsilon _0}ba}}{{b – a}}\)

If \(b \to \infty ,\,C = 4\pi {\varepsilon _0}a,\)

which is the capacitance of a single isolated spherical capacitor of radius \(a.\)

3. **Cylindrical Capacitor:** The cylindrical capacitor comprises a hollow or a solid cylindrical conductor of radius \(a\) surrounded by a similar concentric hollow cylindrical cylinder of radius \(b,\) such that \(a < b.\)

The capacitance \(C\) of a cylindrical capacitor of length \(L\) is

\(C = \frac{{2\pi {\varepsilon _0}L}}{{\ln \frac{b}{a}}}\)

1. **Film Capacitors**: These capacitors use plastic film as a dielectric medium.

2. **Ceramic Capacitors**: These capacitors use ceramic as a dielectric medium.

3. **Electrolytic Capacitors**: These capacitors use an oxide layer as a dielectric medium.

4. **Variable Capacitors**: These capacitors generally use air as a dielectric medium.

The effective capacitance \(C\) of a group of capacitors \({C_1},\,{C_2},{C_3},……{C_n},\) depends on the way the individual capacitors are combined. Two such possible combinations are:

The effective capacitance of a series combination of \(n\) capacitors is,

\(\frac{1}{C} = \frac{1}{{{C_1}}} + \frac{1}{{{C_2}}} + \frac{1}{{{C_3}}}……..\frac{1}{{{C_n}}}\)

The effective capacitance of a parallel combination of \(n\) capacitors is

\(C = {C_1} + {C_2} + {C_3}…..{C_n}\)

*Q.1. Three capacitors, each having a capacitance \(9pF,\) are connected in series, the total capacitance of the combination is? *** Ans:** The capacitance of each capacitor, \(C = 9pF\)

\(\frac{1}{{{C_r}}} = \frac{1}{C} + \frac{1}{C} + \frac{1}{C} + \frac{1}{9} + \frac{1}{9} + \frac{1}{9} = \frac{3}{9}\)

\({C_T} = 3pF\)

*Q.2. Three capacitors of capacitances \(2\,pF,\,3pF\) and \(4\,pF\) are connected in parallel. What is the total capacitance of the combination? *** Ans:** Given, \({C_1} = 2\,pF,{C_2} = \,3pF,{C_3} = 4\,pF\)

The total capacitance, \({C_r} = {C_1} + {C_2} + {C_3} = 2 + 3 + 4 = 9\,pF\)

The energy stored in a capacitor is the electric potential energy. For capacitor having capacitance \(C\) and a potential difference \(V,\) the energy stored in the capacitor will be:

\(E = \frac{1}{2}C{V^2}\)

The energy density of capacitor

\({U_E} = \frac{1}{2}{\varepsilon _0}{E^2}\)

\({\text{E = E}}\) is the electric field

\({\varepsilon _0} = \) permittivity of space

We interpret \({U_E} = \frac{1}{2}{\varepsilon _0}{E^2}\) as the energy density, i.e. the energy per unit volume, in the electric field. The energy stored between the plates of the capacitor equals the energy per unit volume stored in the electric field times the volume between the plates.

(1) The primary function of the capacitor is to store electric charge. Thus, when we want an instantaneous current to flow in a circuit, the best way would be to connect the terminals of the circuit across a capacitor.

(2) A capacitor can be used like a temporary battery to store electrical energy when disconnected from the circuit. In fact when batteries are being replaced, often capacitors are used in electronic devices to maintain the power supply to ensure there is no loss of information in volatile memory.

(3) We use a capacitor in many electrical instruments like ignition systems of motor engines, electric fans, air conditioners etc.

(4) Capacitors pass AC but block DC signals; that is why they are often used to separate the AC and DC components. This method is known as capacitive coupling. Here, an enormous value of capacitance with small reactance is employed.

(5) Groups of large, specially constructed, low-inductance high-voltage capacitors (capacitor banks) are used to supply massive pulses of current for many pulsed power applications. These include electromagnetic forming, pulsed lasers, pulse forming networks, radar, fusion research, and particle accelerators.

(6) Capacitors are used by Dynamic Random Access Memory (DRAM) devices to represent binary information as bits. Capacitors are used along with inductors to tune circuits to particular frequencies, an effect exploited by radio receivers, speakers, and analogue equalizers.

1. The Leyden jar was the first capacitor created, invented in \(1745\) by Pieter van Musschenbroek.

2. Capacitors discharge very slowly, and many of them can store charge for years.

3. Supercapacitors made with graphene as a material for their conducting plates can store charges much like a lithium-ion battery.

4. Capacitors are used in almost all of the devices that we use at home. For example, they are present in Televisions, fans, radios, air conditioners, etc.

**Q. What is the permittivity of space? ****Ans:** The permittivity of space or vacuum is equal to \(1.\).

**Q. What is the SI unit of capacitance?Ans:** The SI unit of capacitance is Farad.

**Q. What kind of energy is stored in a capacitor?****Ans:** Energy stored in a capacitor is electrical potential energy, and it is thus related to the charge QQ and voltage VV on the capacitor.

**Q. What is capacitance?Ans: **Capacitance is the ratio of the amount of electric charge stored on a conductor to the potential difference.

**Q. What is a capacitor?Ans: **A capacitor is a passive device that stores electrical energy in the electric field.

**Q. A metal plate is introduced between the plates of a charged parallel plate capacitor. What is its effect on the capacitance of the capacitor?****Ans:** If a metal plate is introduced between the plates of a charged parallel plate capacitor. The capacitance of the parallel plate capacitor will become infinite.

**Q. Why is capacitance called a passive device?Ans: **Passive devices are devices that do not generate power, and capacitors do not generate power, but they store or dissipate it as any passive device does.

**Q. Why should the electrostatic potential be the same at every point inside a hollow charged conductor?Ans: **There is no work done in moving a small test charge within the conductor since the electric field is zero inside the hollow charged conductor. Therefore, the electrostatic potential inside a hollow charged conductor remains the same at every point.

**Q. In a parallel-plate capacitor, how can the capacitance be decreased? ****Ans:** The capacitance of a parallel plate capacitor is \(C = \frac{{{\varepsilon _0}A}}{d}.\) Thus, the capacitance can be increased by decreasing the distance between the plates.

**Q. Can two equipotential surfaces intersect each other? Ans: **No, two equipotential surfaces cannot intersect each other. If they do, there will be two different directions of electric field at that point which is not correct. If an intersection occurs, then at the same point of intersection, there will be two values of potential. This is not possible and therefore, two equipotential surfaces cannot intersect.