IB Physics Unit 11. Electromagnetic Induction: Capacitance

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:  Cparallel = C1 + C2 + …

➔ series:   1/Cseries   =   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.

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  • 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.

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  • 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.

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CAPACITORS AND CAPACITANCE

A capacitor or condenser is a device that stores electrical energy. It generally consists of two conductors carrying equal but opposite charges.
The ability of a capacitor to hold a charge is measured by a quantity called the capacitance. Let us consider two uncharged identical conductors X and Y and create a P.D. (Potential Difference) V between them by connecting with battery B as shown in figure.
Fig- A capacitor consists of electrically insulated conductors carrying equal positive and negative charge
After connection with the battery, the two conductors X and Y have equal but opposite charges. Such a combination of charged conductors is a device called a capacitor. The P.D. between X and Y is found to be proportional to the charge Q on capacitor.
The capacitance C, of a capacitor is defined as the ratio of the magnitude of the charge on either conductor to the magnitude of P.D. between them.
Capacitance is always a positive quantity.
The S.I. unit of capacitance is coulomb per volt or farad (F).
Furthermore, the value of capacitance depends on size, shape, relative positions of plate, and the medium between the plates. The value of C does not depend on the charge of the plate or p.d. between the plates.

ENERGY STORED IN A CAPACITOR

If Q is charge, V is p.d, C is the capacitance of the capacitor then the energy stored is

SHARING OF CHARGES

When the two charged conductors of capacitances C1 and C2 at potentials V1 and V2 respectively, are connected by a conducting wire, the charge flows from higher to lower potential, until the potentials of the two conductors are equal.
The common potential after sharing of charges,
The charges after sharing on two conductors will be
i.e.,
There is a loss of energy during sharing, converted to heat given by

PARALLEL PLATE CAPACITOR

It consists of two parallel metallic plates of any shape, each of area A and at a distance d apart.
The capacitance of the capacitor is given by

EFFECT OF DIELECTRIC ON CAPACITANCE

When a dielectric slab is placed between the plates of a parallel plate capacitor, the charge induced on its plates due to polarisation of dielectric is
where K = dielectric constant.
When an electric field is applied across a dielectric, induced charges appear on the surface of dielectric which is shown in the above figure. These induced charges produce their own field which acts in the opposite direction of the applied field. Hence, total field is reduced, i.e., E0 = Ep – E where E0 is the applied field, Ep is the induced field and E is the resultant field.
E is given by, where K is the dielectric constant.
If medium between the plates is having a dielectric of dielectric constant K then the capacitance is given
If the space between the plates is partly filled with dielectric then the capacitance of the capacitor will be given by,
,
where t is the thickness of the dielectric with dielectric constant K.
KEEP IN MEMORY
  1. The unit farad is quite a big unit for practical purposes. Even the capacitance of a huge body like earth is 711 μF.
  2. A capacitor is a device which stores charges and produces electricity whenever required.
  3. 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.
  4. A capacitor allows A.C. but doesn’t allow D.C. to pass through it.
  5. The capacitance of a capacitor increases with insertion of a dielectric between its plates and decreases with increase in the separation between the plates.
  6. The capacitance of a capacitor increases K times if a medium of dielectric constant K is inserted between its plates.
  7. 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.
  8. 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

It consists of two concentric spherical conductors of radii R1 and R2. The space between two conductors is filled by a dielectric of dielectric constant K.
  • When outer conductor is earthed,
Capacitance of spherical capacitor,
(without dielectric)
(with dielectric)
  • When inner sphere is earthed,
This is because the combination behaves as two capacitors in parallel, one is a capacitor formed by two concentric spherical shells and the other is an isolated spherical shell of radius R2.

CYLINDRICAL CAPACITOR

It consists of two-coaxial cylindrical conductors of radii R1 and R2, the outer surface of outer conductor being earthed. The space between the two is filled with a dielectric of dielectric constant K.
The capacitance of cylindrical condenser of length l
(without dielectric)
(with dielectric)

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
i.e. Q = C1V1 = C2 V2 = C3 V3
Hence V = V1 + V2 + V3
  • 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
i.e.
Hence, Q = q1 + q2 + q3
  • The equivalent capacitance between a and b is ceq = c1 + c2 + c3 + ……..+ cn
KEEP IN MEMORY
  1. 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
  1. 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
  1. 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

If an electric field E is applied across a parallel plate capacitor filled with a dielectric of dielectric constant K (or permittivity ε), then
Polarisation P = induced charge per unit area (opposite to free charge) =
Electric displacement D = εE = εo E + P
i.e. Polarisation P = (ε – εo) E = (Kεo – εo) E
Electric susceptibility,
Relation between dielectric constant K and electric susceptibility χe

EFFECT OF FILLING DIELECTRIC WITH BATTERY CONNECTED

When there is no dielectric
Capacitance
Potential difference between the plates V
Charge on a plate Q = CV
Energy
Electric field
When dielectric is inserted
Q = K C0 V = KQ0

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

CHARGING AND DISCHARGING A CAPACITOR

CHARGING A CAPACITOR

When an uncharged capacitor is connected across a source of constant potential difference such as a cell, it takes a finite time to get fully charged, although this time interval may be small. This time-interval depends on the capacity of the capacitor and the resistance in the circuit.

DURING THE PERIOD OF CHARGING

  1. The charge on the capacitor increases from ‘zero’ to the final steady charge.
  2. The potential difference developed across the capacitor opposes the constant potential difference of the source.
  3. 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.
  4. As the charge on the capacitor increases, more energy is stored in the capacitor.
  5. 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.
If V0 = constant potential difference of the source
R = pure resistance in the circuit
C = capacity of the capacitor
Q0 = final charge on the capacitor, when fully charged
q = charge on the capacitor at time ‘t’ from the starting of the charging
V = potential difference across the capacitor at time ‘t’
Then
and i = current in the circuit at time ‘t’ =
At time ‘t’ by Kirchhoff’s law

i.e.
Integrating and putting in the initial condition q = 0 at t = 0, we get
Special cases :
  1. At t = 0, q = 0.
  2. When t increases, q increases.
  3. As
  4. At t = CR [‘CR’ has dimensions of time]
This value of t = CR is called the ‘time constant’ of the (CR) circuit.

DISCHARGING OF A CAPACITOR

If after charging the capacitor, the source of constant potential difference is disconnected and the charged capacitor is shorted through a resistance ‘R’, then by Kirchhoff’s law, at time ‘t’ from the instant of shorting,
Putting,
  • the initial condition, q = Q0 at t = 0 and
  • the final condition, q = 0 at ,
the solution to the above equation is
KEEP IN MEMORY
  1. If n small drops each having a charge q, capacity ‘C’ and potential V coalesce to form a big drop, then
    1. the charge on the big drop = nq
    2. capacity of big drop = n1/3 C
    3. potential of big drop = n2/3 V
    4. potential energy of big drop = n5/3 U
    5. surface density of charge on the big drop = n1/3 × surface density of charge on one small drop.
  2. Charged soap bubble : Four types of pressure act on a charged soap bubble.
    1. Pressure due to air outside the bubble PO, acting inwards.
    2. Pressure due to surface tension of soap solution PT, acting inwards.
    3. Pressure due to air inside the bubble, Pi, acting outwards.
    4. Electric pressure due to charging, Pe =, acting outwards.
In equilibrium, Pi + Pe = PO + PT
or, Pi – PO = PT – Pe
or, Pexcess = PT – Pe
Where T = surface tension of soap solution,
σ = surface charge density of bubble.
If Pi = PO then Pi – PO = PT – Pe = 0 or PT = Pe
Hence for maintaining the equilibrium of charged soap bubble,
  1. Force of attraction between the plates of a parallel plate capacitor =
where, A = area of the plates of capacitor, K = dielectric constant of the medium filled between the plates.
In terms of electric field, the force of attraction
  1. Uses of capacitor :
    1. In LC oscillators
    2. As filter circuits
    3. Tuner circuit in radio etc.
  2. 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

METHOD 1 : Successive Reduction
This method is applicable only when the capacitor can be clearly identified as in series or in parallel.
METHOD 2 : Using Symmetry
The above circuit is symmetrical about XAEBY axis. This is because the upper part of the circuit is mirror image of lower part.
Therefore VC = VE = VD. The circuit can be redrawn as
METHOD 3 : Wheatstone bridge
If then the wheatstone bridge is balanced. In this case there will be no charge accumulation in C5 when battery is attached across A and B. Therefore the equivalent circuit is the capacitance C1 and C2 are in series. Similarly C3 and C4 is in series. Therefore the equivalent capacitance occurs between A and B is
The other forms of wheatstone bridge are :
or
METHOD 4 : If none of the above method works, then we can use the method of Kirchhoff’s laws – junction law and loop law.

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