NEET UG - Capacitors and capacitance, combinations in series and parallel, parallel plate capacitor with and without dielectric, energy stored in a capacitor E-Book Study Material

Capacitors and Capacitance

A capacitor is an electrical component that stores electrical energy in an electric field. It consists of two conductive plates separated by an insulating material called a dielectric. When a voltage is applied across the plates, positive charge accumulates on one plate and negative charge on the other. This separation of charge creates an electric field between the plates, and it is in this field that the energy is stored.

Capacitance is a measure of a capacitor's ability to store charge. It is defined as the ratio of the magnitude of the charge on either conductor to the potential difference between them. The unit of capacitance is the Farad (F), named after Michael Faraday. One Farad is a very large unit, so in practice, we often use microfarads (μF, 10^-6 F), nanofarads (nF, 10^-9 F), or picofarads (pF, 10^-12 F).

Mathematically, capacitance (C) is given by the formula:

C = Q / V

Where:

C is the capacitance in Farads (F).
Q is the magnitude of the charge on each plate in Coulombs (C).
V is the potential difference (voltage) across the plates in Volts (V).

A higher capacitance value means that the capacitor can store more charge at a given voltage. The capacitance of a capacitor depends on its physical characteristics, such as the area of the plates, the distance between them, and the type of dielectric material used.

Factors Affecting Capacitance

The capacitance of a parallel plate capacitor is determined by three main factors:

Area of the plates (A): A larger plate area allows for more charge to be stored, thus increasing capacitance.
Distance between the plates (d): A smaller distance between the plates leads to a stronger electric field for a given voltage, allowing more charge to be stored, thus increasing capacitance.
Dielectric material: The insulating material between the plates (dielectric) affects the capacitance. Different materials have different dielectric constants, which influence how much the electric field is reduced when the dielectric is present.

Dielectrics

A dielectric is an electrical insulator that can be polarized by an applied electric field. When a dielectric material is placed between the plates of a capacitor, it reduces the electric field strength between the plates. This reduction in electric field allows more charge to be stored on the plates for the same applied voltage, thereby increasing the capacitance.

The effect of a dielectric is quantified by its dielectric constant (κ), also known as relative permittivity (ε_r). The dielectric constant is a dimensionless quantity that represents the factor by which the capacitance increases when a dielectric material is inserted between the plates compared to when there is a vacuum.

The capacitance of a parallel plate capacitor with a dielectric is given by:

C = κ * C₀

Where:

C is the capacitance with the dielectric.
κ is the dielectric constant of the material.
C₀ is the capacitance with a vacuum between the plates.

For a vacuum, κ = 1. For all other dielectric materials, κ > 1.

Key Concept: A dielectric material enhances the charge storage capacity of a capacitor by reducing the electric field strength between the plates, allowing more charge to accumulate for a given voltage.

Combinations of Capacitors

Capacitors can be combined in series or parallel to achieve desired overall capacitance values. This is similar to how resistors are combined, but the formulas for equivalent capacitance are different.

Capacitors in Series

When capacitors are connected in series, they are connected end-to-end, forming a single path for charge. In a series combination, the charge on each capacitor is the same, but the total voltage across the combination is the sum of the voltages across each individual capacitor.

Consider two capacitors, C₁ and C₂, connected in series with a battery of voltage V. Let Q be the charge on each capacitor, and V₁ and V₂ be the voltages across C₁ and C₂, respectively.

The total voltage is V = V₁ + V₂.

We know that V₁ = Q / C₁ and V₂ = Q / C₂.

Substituting these into the total voltage equation:

V = (Q / C₁) + (Q / C₂)

If we consider an equivalent capacitance C_eq for the series combination, then V = Q / C_eq.

Equating the two expressions for V:

Q / C_eq = (Q / C₁) + (Q / C₂)

Dividing by Q, we get the formula for equivalent capacitance in series:

1 / C_eq = 1 / C₁ + 1 / C₂ + ...

For two capacitors in series, this simplifies to:

C_eq = (C₁ * C₂) / (C₁ + C₂)

Notice that the equivalent capacitance in series is always less than the smallest individual capacitance.

Shortcut for Series Combination (Two Capacitors): Think of it like resistors in parallel, but for capacitance: C_eq = (Product) / (Sum). This is a quick way to calculate the equivalent capacitance for just two capacitors in series.

Capacitors in Parallel

When capacitors are connected in parallel, their terminals are connected together, so the voltage across each capacitor is the same. However, the total charge stored in the combination is the sum of the charges on each individual capacitor.

Consider two capacitors, C₁ and C₂, connected in parallel across a battery of voltage V. Let Q be the total charge, and Q₁ and Q₂ be the charges on C₁ and C₂, respectively.

The total charge is Q = Q₁ + Q₂.

We know that Q₁ = C₁ * V and Q₂ = C₂ * V.

Substituting these into the total charge equation:

Q = (C₁ * V) + (C₂ * V)

If we consider an equivalent capacitance C_eq for the parallel combination, then Q = C_eq * V.

Equating the two expressions for Q:

C_eq * V = (C₁ * V) + (C₂ * V)

Dividing by V, we get the formula for equivalent capacitance in parallel:

C_eq = C₁ + C₂ + ...

The equivalent capacitance in parallel is always greater than the largest individual capacitance.

Shortcut for Parallel Combination: Similar to resistors in series, capacitances in parallel simply add up: C_eq = C₁ + C₂ + ....

Parallel Plate Capacitor

The simplest form of a capacitor is the parallel plate capacitor. It consists of two identical, flat conducting plates of area A, separated by a distance d. The plates are connected to a voltage source V.

Parallel Plate Capacitor with Vacuum

When the space between the plates is a vacuum (or air, approximately), the capacitance (C₀) is given by:

C₀ = (ε₀ * A) / d

Where:

ε₀ is the permittivity of free space, a fundamental constant with a value of approximately 8.854 x 10^-12 F/m.
A is the area of each plate in square meters (m²).
d is the distance between the plates in meters (m).

This formula shows that capacitance is directly proportional to the area of the plates and inversely proportional to the distance between them.

Mnemonic: For a vacuum parallel plate capacitor, remember C = ε₀A/d. Think of ε₀ as the "space's ability" to permit electric fields, A as "how much space" the plates occupy, and d as the "distance" over which the field operates.

Parallel Plate Capacitor with Dielectric

When a dielectric material with dielectric constant κ is inserted to completely fill the space between the plates of a parallel plate capacitor, the capacitance increases. The new capacitance (C) is given by:

C = (κ * ε₀ * A) / d

This can also be written as:

C = κ * C₀

As discussed earlier, the dielectric material reduces the electric field strength between the plates by a factor of κ. For a given voltage V, the charge stored increases because the electric field is weaker.

Example: If a capacitor has a capacitance C₀ in a vacuum, and a dielectric material with κ = 5 is inserted to fill the space, the new capacitance will be 5 times the original capacitance.

Parallel Plate Capacitor with Dielectric Slab of Thickness t

Sometimes, a dielectric slab of thickness t (where t < d) and dielectric constant κ is inserted between the plates. In this case, the space between the plates is partly filled with the dielectric and partly with air (or vacuum).

The total distance between the plates is d. The dielectric slab occupies a thickness t, and the remaining distance (d - t) is filled with air (or vacuum).

The effective capacitance can be calculated by considering the system as two capacitors in series: one with dielectric and thickness t, and another with air/vacuum and thickness (d - t).

The capacitance of the dielectric part (C_dielectric) is:

C_dielectric = (κ * ε₀ * A) / t

The capacitance of the air/vacuum part (C_air) is:

C_air = (ε₀ * A) / (d - t)

Since these are in series, the equivalent capacitance C_eq is given by:

1 / C_eq = 1 / C_dielectric + 1 / C_air

1 / C_eq = t / (κ * ε₀ * A) + (d - t) / (ε₀ * A)

1 / C_eq = [t / κ + (d - t)] / (ε₀ * A)

C_eq = (ε₀ * A) / [t / κ + (d - t)]

This formula shows that the effective distance between the plates is modified. The thickness t of the dielectric is effectively increased by a factor of t/κ in terms of its contribution to the inverse capacitance.

Special Case: If the dielectric slab fills the entire space (t = d), the formula becomes C_eq = (ε₀ * A) / [d / κ + (d - d)] = (ε₀ * A) / (d / κ) = (κ * ε₀ * A) / d, which is the formula for a fully filled dielectric capacitor.

Energy Stored in a Capacitor

When a capacitor is charged, work is done to move charge from one plate to another against the electric field. This work is stored as potential energy in the electric field between the plates.

The process of charging a capacitor can be thought of as moving small amounts of charge (dq) from the negative plate to the positive plate. As charge accumulates on the plates, the voltage across the capacitor increases. The work done to move an infinitesimal charge dq when the voltage is v is dW = v * dq.

Since v = q / C, where q is the instantaneous charge on the capacitor, we have dW = (q / C) * dq.

To find the total energy stored (U) when charging the capacitor from zero charge to a final charge Q, we integrate dW:

U = ∫ dW = ∫₀^Q (q / C) dq

U = (1 / C) * [q² / 2]₀^Q

U = (1 / C) * (Q² / 2)

U = Q² / (2C)

We can also express the energy in terms of voltage. Since Q = C * V, we can substitute this into the energy formula:

U = (C * V)² / (2C) = (C² * V²) / (2C) = (1/2) * C * V²

Alternatively, since C = Q / V, we can substitute this into the (1/2)CV² formula:

U = (1/2) * (Q / V) * V² = (1/2) * Q * V

So, the energy stored in a capacitor can be expressed in three equivalent forms:

U = Q² / (2C)
U = (1/2) * C * V²
U = (1/2) * Q * V

Where:

U is the energy stored in Joules (J).
Q is the charge on the capacitor in Coulombs (C).
C is the capacitance in Farads (F).
V is the voltage across the capacitor in Volts (V).

The formula U = (1/2)CV² is the most commonly used form.

Mnemonic for Energy Stored: Think of charging a capacitor as filling a bucket with water. The 'capacity' is C, the 'height' is V, and the 'amount' is Q. The energy stored is like the potential energy of the water. The formulas are analogous to: U = (amount)² / (2 * capacity), U = (1/2) * capacity * (height)², and U = (1/2) * amount * height. The 1/2 factor arises because the voltage (or height) increases linearly from 0 to its final value during charging.

Energy Density in Dielectric

The energy stored in a capacitor is distributed uniformly throughout the electric field between the plates. The energy density (u) is the energy stored per unit volume.

The volume of the space between the plates of a parallel plate capacitor is Volume = A * d.

Using the formula U = (1/2)CV² and C = (κ * ε₀ * A) / d, we get:

U = (1/2) * [(κ * ε₀ * A) / d] * V²

The electric field strength (E) in a parallel plate capacitor filled with a dielectric is E = V / d, so V = E * d.

Substituting V:

U = (1/2) * [(κ * ε₀ * A) / d] * (E * d)²

U = (1/2) * [(κ * ε₀ * A) / d] * E² * d²

U = (1/2) * κ * ε₀ * A * E² * d

Now, energy density (u) is U divided by the volume (A*d):

u = U / (A * d) = [(1/2) * κ * ε₀ * A * E² * d] / (A * d)

u = (1/2) * κ * ε₀ * E²

This formula gives the energy density stored in the electric field within the dielectric material. For a vacuum (κ=1), the energy density is u₀ = (1/2) * ε₀ * E².

Important Note: The energy stored in a capacitor is not actually located on the plates, but rather in the electric field in the region between the plates.