A cylindrical capacitor is a specific type of capacitor characterized by its cylindrical structure. It consists of two coaxial (aligned along the same axis) cylinders or conductors, one inside the other, separated by a dielectric material. Here are the key components and characteristics of a cylindrical capacitor: 1. **Structure**: It comprises an inner cylinder and an outer cylinder, both arranged along the same axis. The space between these cylinders is filled with a dielectric material that prevents direct electrical contact between the cylinders. 2. **Dielectric Medium**: The dielectric material, which could be air, vacuum, or other non-conductive substances, helps in maintaining a potential difference between the cylinders without allowing the flow of current between them. 3. **Capacitance Factors**: The capacitance of a cylindrical capacitor is influenced by several factors, including the radii of the cylinders, their lengths, and the properties of the dielectric material between them. Formulas exist to calculate the capacitance based on these parameters. 4. **Applications**: Cylindrical capacitors find applications in various fields such as electronics, power systems, and telecommunications due to their relatively high capacitance compared to other capacitor designs. They are utilized where efficient energy storage in a compact form is required. 5. **Energy Storage**: During charging, energy from an external source is expended to charge the capacitor. This energy gets stored within the electrostatic field formed in the dielectric material. Upon discharge, this stored energy is released. 6. **Functions**: Cylindrical capacitors serve multiple functions, including energy storage, signal processing in circuits, filtering, and regulation of electrical energy in power transmission systems. In summary, cylindrical capacitors are a specific design of capacitors consisting of coaxial cylinders separated by a dielectric medium. Their structure and properties make them valuable in various technological applications where efficient energy storage and manipulation of electrical signals are required.

1. Cylindrical
Capacitor
• Basic Structure: A cylindrical capacitor comprises two
coaxial cylinders, typically an inner and an outer cylinder,
aligned along the same axis.
• Dielectric Medium: It involves an insulating material, the
dielectric, placed between the cylinders to prevent direct
electrical contact. This dielectric can vary from air to specific
non-conductive substances.
• Capacitance Factors: The capacitance of a cylindrical
capacitor depends on the radii of the cylinders, their lengths,
and the permittivity of the dielectric material. Formulas exist
to calculate capacitance based on these parameters.
• Applications: Cylindrical capacitors are used widely in
electronics, power systems, and telecommunications. They're
valued for their relatively high capacitance compared to
other capacitor designs.
• Practical Use: They serve various functions, including
energy storage, signal processing in circuits, filtering, and
regulation of electrical energy in power transmission systems.

2. 1. A single-core cable or cylindrical capacitor consisting
two co-axial cylinders of radii a and b metre.
2. Let the charge per metre length of the cable on the
outer surface of the inner cylinder be + Q coulomb
and on the inner surface of the outer cylinder be −Q
coulomb.
3. Let εr be the relative permittivity of the medium
between the two cylinders. The outer cylinder is
earthed.
4. let us find the value of electric intensity at any point
distant x metres from the axis of the inner cylinder.
5. Total flux coming out radially from the curved surface
of this imaginary cylinder is Q coulomb. Area of the
curved surface = 2 π x × 1 = 2 π x m2 .

5. Potential Gradient in a Cylindrical Capacitor

9. Capacitance Between
Two Parallel Wires
• This case is of practical importance in overhead
transmission lines. The simplest system is 2-wire system (either
d.c. or a.c.). In the case of a.c. system, if the transmission line
is long and voltage high, the charging current drawn by the line
due to the capacitance between conductors is appreciable and
affects its performance considerably.
• let d = distance between centres of the wires A and B
• r = radius of each wire (≤d)
• Q = charge in coulomb/metre of each wire
• Now, let us consider electric intensity at any point P
between conductors A and B. Electric intensity at P* due to
charge + Q coulomb/metre on A is

12. Capacitors
in Series
1. C1 , C2 , C3 = Capacitances
of three capacitors
2. V1 , V2 , V3 = p.ds. across
three capacitors.
3. V = applied voltage across
combination
4. C = combined or equivalent
or joining capacitance.
In series combination, charge on
all capacitors is the same but
p.d. across each is different.

13. Capacitors in
Parallel
In this case, p.d. across each is the
same but charge on each is different.
∴ Q = Q1 + Q2 + Q3 or CV = C1V + C2V
+ C3V or C = C1 + C2 + C3 For such a
combination, dV/dt is the same for all
capacitors.

14. Cylindrical Capacitor with Compound
Dielectric

17. Insulation
Resistance of a
Cable Capacitor

18. • Current Flow in Cable Capacitor: Current primarily flows along the axis of the
core in a cable capacitor, serving its intended purpose.
• Leakage of Current: However, some current inevitably leaks, and this leakage
occurs radially, perpendicular to the intended current flow.
• Insulation Resistance: The resistance presented to this radial leakage current
is termed as the insulation resistance of the cable.
• Relation to Cable Length: With longer cable lengths, there's an increase in
leakage due to the radial current flow, resulting in reduced insulation
resistance. In essence, as cable length grows, more current leaks, leading to a
decrease in insulation resistance.
• Inverse Proportionality: The relationship between insulation resistance and
cable length is inversely proportional. Longer cables exhibit lower insulation
resistance due to increased leakage.
• Differentiation from Conductor Resistance: It's crucial not to confuse
insulation resistance with conductor resistance. Insulation resistance pertains
to the resistance against radial leakage and varies with cable length, whereas
conductor resistance, related to the conductor material, is directly
proportional to the cable length.

21. Energy Stored in a
Capacitor
• Initial Energy Expenditure: Charging a capacitor requires
energy from an external source. This energy is stored within
the electrostatic field that forms in the dielectric medium.
• Energy Storage and Discharge: The energy expended during
charging gets stored in the electrostatic field. Upon
discharge, this stored energy is released as the field
collapses.
• Work During Charging: Initially, when the capacitor is
uncharged, minimal work is needed to transfer charge
between the plates. However, as more charge is added,
overcoming the repulsive force between the accumulated
charges demands additional work.
• Calculation of Energy Spent: The energy spent in charging a
capacitor of capacitance C to a voltage V can be determined.
At any point during charging, if the potential difference
across the plates is v and the next charge increment
transferred is 'dq', the work done is involved in moving 'dq'
charge from one plate to the other.

24. Force of Attraction Between Oppositely-
charged Plates
1. Two parallel conducting plates A and B carrying
constant charges of + Q and −Q coulombs
respectively. Let the force of attraction between the
two be F newtons. If one of the plates is pulled apart
by distance dx,
then work done is = F × dx joules ...(i)
2. Since the plate charges remain constant, no
electrical energy comes into the arrangement during
the movement dx.
3. ∴ Work done = change in stored energy

27. Tutorial-5.2

28. Current-Voltage Relationships in a Capacitor

31. Time Constant

32. Discharging of a Capacitor
• when S is shifted to b, C is discharged through R. It will be seen that
the discharging current flows in a direction opposite to that the
charging current.
• Hence, if the direction of the charging current is taken positive, then
that of the discharging current will be taken as negative. To begin
with, the discharge current is maximum but then decreases
exponentially till it ceases when capacitor is fully discharged.

36. Transient Relations During Capacitor Charging Cycle
1. Steady-State Transition: When a circuit moves from one stable state to another, it goes
through a transient state. This transient state is brief compared to the overall duration.
2. Initial and Final Conditions: The initial state is the starting stable condition, and the final
state is the target stable condition of the circuit.
3. Transient Phase: The transient state occurs between the initial and final conditions. It
represents the period during which changes in current and voltage occur in the circuit.
4. Example with RC Circuit: Consider an RC circuit where initially, the switch isn't
connected (at neither a nor b). This is the initial steady state with no current flow and
consequently no voltage drops across components.
5. Switch Position Change: When the switch moves to position a, current flows through
resistor R, initiating changes in voltage across R and the capacitor C. During this phase,
transient voltages develop until they reach their stable, final values.
6. Duration of Transient Condition: The time frame in which current and voltage variations
occur and stabilize is termed as the transient condition in the circuit.
In essence, this process describes how a circuit, such as an RC circuit, undergoes changes in
current and voltage as it transitions from an initial stable state to a final stable state, passing
through a brief transient state in between.

38. Transient Relations During Capacitor Discharging Cycle

39. Charging and Discharging of a capacitor with Initial Charge
• Initial Capacitor Potential: In this scenario, the capacitor starts with an
initial potential (Vo) that is lower than the applied battery voltage (V).
• Opposition to Applied Voltage: The initial potential (Vo) acts in opposition
to the battery voltage (V) during the charging process.
• Effect on Rate of Rise: illustrates that the initial rate of rise of the capacitor
voltage (vc ) is somewhat slower when the capacitor begins with an initial
potential (V) compared to when it starts uncharged.
• Rise from V to Final V: The capacitor's voltage rises from the initial value
(V0 ) to the final value (V) over one time constant.
• Calculation of Initial Rate of Rise: The initial rate of rise of the capacitor
voltage (vc ) from an initial value of V0 to the final value of V in one time
constant can be determined based on the specific circuit characteristics
and the relationship between the initial and final voltages.

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