Electric Potential and Capacitance Class 12 Physics Chapter 2 Notes

October 2, 2023 / By / CBSE Class 12 Physics Notes

Electric Potential and Capacitance Class 12 Physics Chapter 2 Notes

Electrostatic Potential

  1. Introduction to Electrostatic Potential
    • Electrostatic potential is defined for a general static charge configuration.
    • It represents the potential energy of a test charge q in terms of the work done on it.
    • The work done is proportional to q and is divided by q to make it independent of the test charge’s magnitude.
  2. Work Done per Unit Test Charge
    • The work done per unit test charge is characteristic of the electric field associated with the charge configuration.
    • This leads to the concept of electrostatic potential V, which is defined as the electric potential energy per unit charge.
    • Work done by an external force in bringing a unit positive charge from point R to P: VP – VR = U_P – U_R = -∆U / q
    • VP and VR are the electrostatic potentials at points P and R, respectively.
  3. Potential Difference is Significant
    • Like potential energy, it’s the potential difference that is physically meaningful.
    • Choosing the potential to be zero at infinity simplifies calculations.
    • Work done by an external force in bringing a unit positive charge from infinity to a point equals the electrostatic potential (V) at that point.
  4. Calculating Potential
    • To determine the work done per unit test charge (dW/dq), an infinitesimal test charge dq is used.
    • The work done (dW) in bringing dq from infinity to the point is divided by dq.
    • An external force at each point of the path counteracts the electrostatic force on the test charge.

Potential Due to a Point Charge

  1. Consider a Point Charge
    • A point charge Q is placed at the origin.
    • We want to find the electric potential at any point P with position vector r from the origin.
  2. Determining the Potential
    • To find the potential at point P, we calculate the work done in bringing a unit positive test charge from infinity to point P.
    • The work is done against the repulsive force from the charge Q, which is positive for Q > 0.
    • We choose the path along the radial direction from infinity to point P for convenience.
  3. Electrostatic Force along the Path
    • At an intermediate point P’ on the path, the electrostatic force on a unit positive charge is given by: F = (1 / (4πε₀)) * (Q / r²) * r̂’
      • r̂’ is the unit vector along OP’.
    • Work done against this force from r’ to r’ + Δr’ is: ΔW = – (1 / (4πε₀)) * (Q² / r’) * Δr’
  4. Total Work Done
    • To find the total work done (W), we integrate ΔW from r’ = ∞ to r’ = r: W = ∫[∞ to r] (- (1 / (4πε₀)) * (Q² / r’) * Δr’) = (-Q / (4πε₀)) * ∫[∞ to r] (1 / r’) * Δr’
  5. Electric Potential
    • By definition, the potential (V) at point P due to charge Q is:
    • V = -W / q = (Q / (4πε₀)) * ∫[∞ to r] (1 / r’) * Δr’ = (Q / (4πε₀)) * [1/r] = (1 / (4πε₀)) * (Q / r)
  6. Sign Convention
    • Equation (above) is true for any sign of charge Q, though it’s derived for Q > 0.
    • For Q < 0, potential (V) is negative, indicating an attractive force on a unit positive test charge.
  7. Potential at Infinity
    • The choice that the potential at infinity is zero is consistent with Eq. (2.8).
  8. Variation of Electrostatic Potential and Field
    • The electrostatic potential (proportional to 1/r) and the electrostatic field (proportional to 1/r²) vary with distance r from the point charge.

Potential Due to an Electric Dipole

  • An electric dipole consists of two charges, q and -q, separated by a small distance 2a, with a total charge of zero.
  • It is characterized by a dipole moment vector, p, with magnitude q × 2a, pointing from -q to q.
  • The electric field of a dipole depends on both the magnitude of the position vector r and the angle between r and p. It falls off as 1/r^3 at large distances.

Electric Potential Due to a Dipole

  • Electric potential follows the superposition principle. The potential due to a dipole is the sum of potentials due to its charges q and -q.
  • The potential due to a dipole at a point P with position vector r is given by:
    • V = (1/4πε₀) * [(q/|r₁|) – (q/|r₂|)]
    • Where r₁ and r₂ are distances from P to q and -q, respectively.

Approximations for r >> a (Distance Much Greater Than Dipole Size)

  • Using geometry, we approximate r₁ and r₂:
    • r₁ ≈ r – (a cosθ)
    • r₂ ≈ r + (a cosθ)
    • Where θ is the angle between r and p.
  • Applying the binomial theorem and retaining terms up to the first order in a/r, we obtain:
    • (1/|r₁|) ≈ (1/r) + (a/r³) cosθ
    • (1/|r₂|) ≈ (1/r) – (a/r³) cosθ
  • Substituting these approximations into the potential equation and using p = 2qa, we get:
    • V = (1/4πε₀) * (2qa cosθ / r³)
    • V = (1/4πε₀) * (p cosθ / r³)

Electric Potential of a Dipole

  • The electric potential of a dipole is given by:
    • V = (1/4πε₀) * (p⋅r̂ / r³)
    • Where r̂ is the unit vector along the position vector OP.
  • This equation is approximately true only for distances much greater than the dipole size (r >> a). For a point dipole at the origin, it is exact.

Potential Due to a System of Charges

  • Consider a system of charges q₁, q₂, …, qₙ with position vectors r₁, r₂, …, rₙ relative to some origin.
  • The potential V₁ at point P due to charge q₁ is given by:
    • V₁ = (1/4πε₀) * (q₁ / r₁P)
    • Where r₁P is the distance between q₁ and P.
  • Similarly, the potentials V₂ and V₃ at point P due to charges q₂ and q₃ are given by:
    • V₂ = (1/4πε₀) * (q₂ / r₂P)
    • V₃ = (1/4πε₀) * (q₃ / r₃P)
  • By the superposition principle, the potential V at point P due to the total charge configuration is the sum of potentials due to individual charges:
    • V = V₁ + V₂ + … + Vₙ
    • V = (1/4πε₀) * (q₁/r₁P + q₂/r₂P + … + qₙ/rₙP)

Continuous Charge Distribution

  • For a continuous charge distribution with charge density ρ(r), we divide it into small volume elements, each carrying charge ρ(r)dV.
  • We determine the potential due to each volume element and integrate over all such contributions to find the potential due to the entire distribution.

Potential of a Uniformly Charged Spherical Shell

  • For a uniformly charged spherical shell with radius R and total charge q, the electric field outside the shell is as if the entire charge is concentrated at the center.
  • The potential outside the shell (R ≥ r) is given by:
    • V = (1/4πε₀) * (q / r)
  • Inside the shell, the electric field is zero, implying that potential is constant (no work is done in moving a charge inside the shell), and equals its value at the surface:
    • V = (1/4πε₀) * (q / R)

Equipotential Surfaces

  • An equipotential surface is a surface where the potential has a constant value at all points on that surface.
  • For a single charge q, the potential is given by V = (1/4πε₀) * (q / r), which implies that V is constant when r is constant.
  • Equipotential surfaces for a single point charge are concentric spherical surfaces centered at the charge.

Electric Field and Equipotential Surfaces

  • Electric field lines for a single charge q are radial lines starting from or ending at the charge, depending on its polarity (positive or negative).
  • The electric field at any point on an equipotential surface is normal (perpendicular) to that surface.
  • This holds true for any charge configuration: the equipotential surface through a point is always normal to the electric field at that point.
  • Proof: If the field were not normal to the equipotential surface, it would have a non-zero component along the surface. To move a unit test charge against this component would require work, contradicting the definition of an equipotential surface (no potential difference between points). Therefore, the electric field must be normal to the equipotential surface at every point.

Visualization and Alternative Perspective

  • Equipotential surfaces provide an alternative visual representation to electric field lines in understanding the electric field around a charge configuration.
  • They show where the potential is constant in space, providing a different way to visualize the distribution of electric potential.

Uniform Electric Field and Equipotential Surfaces

  • In the case of a uniform electric field E, such as one along the x-axis, the equipotential surfaces are planes that are normal (perpendicular) to the direction of the field.
  • In this example, the equipotential surfaces are planes parallel to the y-z plane, indicating that the potential remains constant along lines perpendicular to the field direction.

Equipotential surfaces are a useful concept for understanding and visualizing electric fields and potentials in various charge configurations. They are particularly valuable in situations where the electric field may not be as intuitive to visualize directly.

Relation between Electric Field and Electric Potential

Consider two closely spaced equipotential surfaces A and B with potential values V and V + dV, where dV is the change in V in the direction of the electric field E. Let P be a point on surface B, and dl be the perpendicular distance of surface A from P.

  1. Work Done in Moving a Unit Positive Charge:
    • Imagine moving a unit positive charge from surface B to surface A against the electric field E. The work done in this process is equal to the potential difference VA – VB.
    • Work done = |E| * dl
    • VA – VB = |E| * dl
  2. Magnitude of Electric Field:
    • Rewriting the equation as |E| = -dV / dl (since dV is negative):
    • |E| = δV / δl
    • |E| represents the magnitude of the electric field.
    • This equation tells us that the magnitude of the electric field is equal to the rate of change of the electric potential per unit displacement normal (perpendicular) to the equipotential surface at the point.

Conclusions about the Relationship between Electric Field and Electric Potential:

  1. Direction of the Electric Field:
    • The electric field is in the direction in which the electric potential decreases most rapidly. In other words, it points from higher potential to lower potential.
  2. Magnitude of the Electric Field:
    • The magnitude of the electric field is determined by the rate of change of the electric potential per unit displacement normal to the equipotential surface at a specific point.
    • Essentially, it quantifies how rapidly the potential changes as one moves perpendicular to an equipotential surface.

These conclusions are fundamental for understanding the interplay between electric fields and electric potentials and provide insights into the behavior of electric fields in relation to potential surfaces.

Potential Energy of a System of Charges

Let’s consider the potential energy of a system of charges, starting with two charges q₁ and q₂ located at r₁ and r₂, respectively.

  1. Work Done in Building Up the Configuration:
    • Initially, both charges q₁ and q₂ are at infinity.
    • Bringing q₁ from infinity to r₁ requires no external work because there’s no opposing field.
    • The potential at a point P due to q₁ is given by V₁ = (1/4πε₀) * (q₁ / r₁P).
  2. Work Done in Bringing q₂ from Infinity to r₂:
    • The work done to bring q₂ to r₂ is given by q₂ * V₂, where V₂ is the potential at r₂ due to q₁.
    • Work done on q₂ = q₂ * V₂ = (q₂ / 4πε₀) * (q₁ / r₂1).
  3. Potential Energy of the System:
    • Since the electrostatic force is conservative, this work is stored as potential energy (U) of the system.
    • U = q₁ * V₁ + q₂ * V₂
    • U = (1/4πε₀) * (q₁ * q₂ / r₂1).
    This expression for potential energy is independent of the order in which the charges are brought to their positions due to the path-independence of work in electrostatic forces.

Significance of Potential Energy:

  • The sign of the potential energy depends on the nature of the charges:
    • For like charges (q₁q₂ > 0), the electrostatic force is repulsive. Therefore, a positive amount of work is needed to separate them from infinity, resulting in positive potential energy.
    • For unlike charges (q₁q₂ < 0), the electrostatic force is attractive. Work is done to separate them from infinity, and this results in negative potential energy.

Generalization for Multiple Charges:

  • This potential energy expression can be generalized for a system of any number of point charges.
  • For example, for three charges q₁, q₂, and q₃ located at r₁, r₂, and r₃, respectively:
    • The total potential energy U is given by the sum of the work done to assemble each charge in the system.
    U = (1/4πε₀) * [(q₁q₂ / r₂1) + (q₁q₃ / r₃1) + (q₂q₃ / r₃2)].
  • The final expression for U is independent of the manner in which the charges are assembled, as the electrostatic force is conservative.

Potential Energy of a Charge in an External Field

In this section, we discuss the concept of potential energy for a charge q in the presence of an external electric field, E, created by sources external to the charge q.

  1. External Electric Field E:
    • The electric field E at a given point is produced by external sources, and the nature of these sources may be known or unknown.
    • We assume that the charge q does not significantly affect the sources creating the external field. This holds true if q is very small or if the external sources are held fixed by other unspecified forces.
    • We are interested in determining the potential energy of charge q (and later, systems of charges) within this external field, not the potential energy of the sources producing the external field.
  2. Definition of Potential Energy:
    • The potential energy of charge q in the external field is given by:
      • Potential energy of q at position r in an external field = qV(r) (Equation 2.27)
    • Here, V(r) is the external potential at the point with position vector r relative to some origin.
  3. Work Done to Bring Charge q:
    • The potential energy is related to the work done in bringing a charge q from infinity to a specific point P in the external field.
    • Work done = qV, where V is the potential at point P.
    • The work done is stored as potential energy of q.
  4. Unit of Energy – Electron Volt (eV):
    • If an electron with charge q = e = 1.6×10⁻¹⁹ C is accelerated by a potential difference of DV = 1 volt, it gains an energy of qDV = 1.6 × 10⁻¹⁹ J.
    • This energy unit is defined as 1 electron volt (eV), and it is commonly used in atomic, nuclear, and particle physics:
      • 1 eV = 1.6 × 10⁻¹⁹ J
      • Other common energy units based on eV include:
        • 1 keV = 10³ eV = 1.6 × 10⁻¹⁶ J
        • 1 MeV = 10⁶ eV = 1.6 × 10⁻¹³ J
        • 1 GeV = 10⁹ eV = 1.6 × 10⁻¹⁰ J
        • 1 TeV = 10¹² eV = 1.6 × 10⁻⁷ J

Potential Energy of a System of Two Charges in an External Field

Here, we calculate the potential energy of a system of two charges, q₁ and q₂, located at positions r₁ and r₂, respectively, within an external electric field E.

  1. Work Done to Bring q₁ from Infinity to r₁:
    • The work done in bringing charge q₁ from infinity to position r₁ is given by q₁V(r₁).
  2. Work Done to Bring q₂ from Infinity to r₂:
    • The work done in bringing charge q₂ from infinity to position r₂ consists of two parts:
      • Work done against the external field E: q₂V(r₂).
      • Work done against the electric field due to q₁, which is the field produced by q₁ and affects the motion of q₂. This work is given by (q₂ / 4πε₀) * (q₁ / r₁₂).
  3. Total Work Done in Assembling the Configuration:
    • By the superposition principle for fields, we add up the work done on q₂ against both the external field and the field due to q₁.
    • Total work done to bring q₂ to r₂:
      • W = q₂V(r₂) + (q₂ / 4πε₀) * (q₁ / r₁₂)
  4. Total Potential Energy of the System:
    • The potential energy of the system, U, is equal to the total work done in assembling the configuration of charges q₁ and q₂.
    • U = q₁V(r₁) + q₂V(r₂) + (q₂ / 4πε₀) * (q₁ / r₁₂).

This expression represents the potential energy of a system consisting of two charges q₁ and q₂ in an external electric field E. It takes into account the work done to bring each charge from infinity to its respective position and the work done against both the external field and the field due to the other charge.

Potential Energy of a Dipole in an External Electric Field

Here, we calculate the potential energy of a dipole consisting of two charges, q₁ = +q and q₂ = -q, placed in a uniform electric field E. The dipole experiences no net force but a torque due to the electric field, which tends to rotate it. We consider an external torque, τ_ext, applied to the dipole to neutralize this torque and rotate it from an initial angle θ₀ to a final angle θ₁ at an infinitesimal angular speed without angular acceleration.

  1. Work Done by External Torque:
    • The work done by the external torque τ_ext is given by:
      • Work = τ_ext * dθ = pE * (cos(θ₁) – cos(θ₀)), where p is the dipole moment vector and E is the uniform electric field.
  2. Potential Energy of the Dipole:
    • The work done by the external torque is stored as the potential energy of the dipole.
    • We define potential energy U(q) with respect to the inclination angle θ of the dipole. A common choice is to set θ₀ = π/2, which results in U(q) = -pE cos(θ).
    • This choice ensures that when θ = π/2 (dipole aligned perpendicular to the field), the potential energy is zero.
  3. Alternative Expression for Potential Energy:
    • The potential energy U(q) applied to the system of charges +q and -q. It leads to the expression:
      • U(q) = (1/4πε₀) * (q² / a) * [V(r₁) – V(r₂)], where r₁ and r₂ are the position vectors of +q and -q, and a is the separation distance between the charges.
  4. Simplifying the Expression:
    • We recognize that the potential difference between positions r₁ and r₂ is equal to the work done in bringing a unit positive charge against the electric field from r₂ to r₁, which is -E * 2a cos(θ).
    • This allows us to simplify the expression further to:
      • U(q) = -pE * cos(θ).
  5. Choice of θ₀ = π/2:
    • The choice θ₀ = π/2 ensures that the work done against the external field in bringing +q and -q to their respective positions is equal and opposite, resulting in a net work of zero.
    • This choice simplifies calculations and aligns with common practice.

Electrostatics of Conductors

In this section, we discuss various important electrostatic properties of conductors:

  1. Electric Field Inside a Conductor Is Zero:
    • In a static situation, when there is no current inside or on the surface of a conductor, the electric field is zero everywhere inside the conductor.
    • Conductors have free charge carriers (e.g., electrons), and as long as the electric field is not zero, these charge carriers would experience forces and drift.
    • In the static case, the free charges distribute themselves such that the electric field is zero inside, which is a defining property of conductors.
  2. Electric Field at the Surface of a Charged Conductor:
    • The electric field at the surface of a charged conductor must be normal (perpendicular) to the surface at every point.
    • If the electric field had tangential components, free charges on the surface would experience forces and move.
    • This ensures that in the static situation, the electric field has no tangential component at the surface.
  3. No Excess Charge Inside a Conductor in the Static Situation:
    • In a neutral conductor, equal amounts of positive and negative charges exist in every small volume or surface element.
    • When the conductor is charged, excess charge can reside only on the surface in the static situation.
    • This is consistent with Gauss’s law, which states that the net charge enclosed by a closed surface is zero.
  4. Constant Electrostatic Potential Throughout the Conductor:
    • In a conductor, the electrostatic potential is constant throughout its volume.
    • There is no potential difference between any two points inside or on the surface of the conductor.
    • The potential inside the conductor has the same value as inside it.
  5. Electric Field at the Surface of a Charged Conductor (for S > 0):
    • The electric field at the surface of a charged conductor with a surface charge density s > 0 is given by:
      • E = s / (ε₀), where ε₀ is the permittivity of free space.
    • The electric field points outward from the conductor’s surface.
  6. Electrostatic Shielding:
    • Electric field inside a conductor’s cavity is always zero, regardless of the cavity’s size, shape, the conductor’s charge, or external fields.
    • Even if the conductor is charged or induced by an external field, all charges are located only on the outer surface of the conductor.
    • This property is known as electrostatic shielding and can protect sensitive instruments from external electrical influence.

Dielectrics and Polarization

Dielectrics are non-conducting substances that do not have a significant number of charge carriers, unlike conductors. In conductors, free charge carriers move to adjust the charge distribution in response to an external electric field, eventually canceling out the field. In dielectrics, free charge movement is not possible, but an external electric field can induce a dipole moment in the molecules of the dielectric. This induced dipole moment results from the stretching or reorientation of molecules.

The charge distribution at the molecular level within a dielectric:

  1. Polar and Non-Polar Molecules:
    • Non-polar molecules have centers of positive and negative charges that coincide, resulting in no permanent dipole moment.
    • Examples include oxygen (O₂) and hydrogen (H₂) molecules.
    • Polar molecules have separated positive and negative charge centers, creating a permanent dipole moment even without an external field.
    • Examples include HCl or water (H₂O) molecules.
  2. Polarization of Non-Polar Molecules:
    • In an external electric field, the positive and negative charges of non-polar molecules are displaced in opposite directions.
    • The molecules develop an induced dipole moment as the external force on charges is balanced by restoring forces.
    • This process is known as polarization, where the dielectric develops an induced dipole moment in the direction of the field, proportionate to the field strength.
  3. Polarization of Polar Molecules:
    • Polar molecules, even without an external field, have randomly oriented permanent dipoles due to thermal agitation.
    • When an external field is applied, these dipoles tend to align with the field, resulting in polarization.
    • The extent of polarization depends on the balance between dipole potential energy aligning dipoles with the field and thermal energy disrupting the alignment.
  4. Polarization as Dipole Moment per Unit Volume (P):
    • Polarization is defined as the dipole moment per unit volume.
    • For linear isotropic dielectrics, the relationship between polarization (P), electric field (E), and electric susceptibility (χe) is given by: P = χeε₀E, where ε₀ is the permittivity of free space.

When a dielectric is placed in an external electric field, it modifies the field within it. The polarization of the dielectric causes induced charges to accumulate at its surfaces. This surface charge density opposes the external field, reducing the total electric field within the dielectric. The induced charges originate from the bound charges within the dielectric, not from free charges.

Capacitors and Capacitance

A capacitor is a device consisting of two conductors separated by an insulating material, often called a dielectric. The two conductors are typically referred to as plates. When charges of equal magnitude but opposite sign are placed on the plates, they create an electric field between them. Capacitors are used to store electric energy in the form of electrostatic potential energy.

Here are the key points about capacitors and capacitance:

  1. Capacitor Configuration:
    • A typical capacitor consists of two conductors (plates) separated by an insulator (dielectric).
    • The conductors can have charges, Q₁ and Q₂, and potentials, V₁ and V₂.
    • In most practical cases, one conductor has a charge of Q, while the other has a charge of -Q, resulting in a total charge of zero for the capacitor.
    • The potential difference (voltage) across the capacitor is given by V = V₁ – V₂.
  2. Capacitance (C):
    • The capacitance of a capacitor is represented by the symbol C.
    • It is defined as the ratio of the charge (Q) on one of the conductors to the potential difference (V) between the conductors: C = Q / V
    • Capacitance is a constant that depends solely on the physical characteristics (geometry and dielectric material) of the capacitor and is independent of the charge or voltage.
    • The SI unit of capacitance is the farad (F), where 1 F = 1 C/V.
  3. Relation Between Charge, Voltage, and Capacitance:
    • The relationship between charge (Q), voltage (V), and capacitance (C) is given by: Q = C * V
    • This equation shows that for a given capacitance, the charge stored on the capacitor is directly proportional to the voltage across it.
  4. Dielectric Strength:
    • The dielectric strength of a dielectric material is the maximum electric field it can withstand without losing its insulating properties.
    • For air, the dielectric strength is approximately 3 × 10^6 V/m, which corresponds to a potential difference of 3 × 10^4 V between the conductors for a typical separation of about 1 cm.
    • Capacitors with high capacitance can store larger amounts of charge at lower voltages, reducing the risk of dielectric breakdown.
  5. Practical Units:
    • The farad (F) is a large unit of capacitance. More commonly used units include the millifarad (mF = 10⁻⁶ F), microfarad (μF = 10⁻⁹ F), nanofarad (nF = 10⁻¹² F), and picofarad (pF = 10⁻¹⁵ F).
    • These smaller units are more convenient for describing capacitors in electronic circuits.

The Parallel Plate Capacitor

A parallel plate capacitor is a simple and commonly used type of capacitor in which two large, flat conducting plates are placed parallel to each other and separated by a small distance. Let’s discuss the key characteristics of a parallel plate capacitor:

  1. Plate Area (A) and Separation (d):
    • The capacitor consists of two conducting plates with an area A.
    • The plates are separated by a distance d, which is much smaller than the dimensions of the plates (d² << A). This assumption is made for simplicity in calculations.
  2. Plate Charges:
    • One plate is positively charged with a charge Q, and the other plate is negatively charged with a charge -Q. The total charge on the capacitor is zero (Q – Q = 0).
  3. Electric Field Between Plates:
    • The electric field inside the capacitor between the plates is uniform and directed from the positively charged plate to the negatively charged plate.
    • The magnitude of the electric field (E) is the same throughout this region.
  4. Electric Field Outside Plates:
    • Outside the plates, the electric field is zero, as there are no charges present in these regions.
  5. Surface Charge Density (σ):
    • Each plate has a surface charge density, σ, defined as the charge per unit area (σ = Q/A).
    • One plate has a positive surface charge density, and the other has a negative surface charge density of the same magnitude.
  6. Electric Potential Difference (Voltage):
    • The electric potential difference (voltage), V, between the two plates is the work done per unit positive charge in moving a charge from the negatively charged plate to the positively charged plate.
    • The voltage across the capacitor is proportional to the charge on the plates and inversely proportional to the distance between the plates: V = Q / (ε₀A) * d where ε₀ is the vacuum permittivity constant.
  7. Capacitance (C):
    • The capacitance (C) of a parallel plate capacitor is defined as the ratio of the charge on one of the plates (Q) to the voltage across the plates (V): C = Q / V = ε₀A / d
    • The capacitance depends only on the physical characteristics of the capacitor, such as the area of the plates and the separation between them.
    • The SI unit of capacitance is the farad (F), where 1 F = 1 C/V.
  8. Significance of Capacitance:
    • A larger capacitance means that the capacitor can store more charge for a given voltage. In practical terms, this means that larger capacitors can store more energy.
    • Capacitors with smaller capacitance values are often used in electronic circuits for tasks such as filtering, timing, and energy storage.
  9. Effect of Dielectric:
    • The effect of placing a dielectric material between the plates of a parallel plate capacitor is discussed in the next section. Dielectrics increase the capacitance of the capacitor.
  10. Magnitude of 1 Farad (F):
    • One farad (1 F) is a relatively large unit of capacitance. For typical values of plate area (A) and separation (d), the vacuum permittivity constant (ε₀) is approximately 8.85 x 10⁻¹² F/m.
    • To visualize the magnitude of 1 F, consider that for plates separated by 1 cm (0.01 m), the required area of each plate to achieve a capacitance of 1 F is roughly 30 square kilometers.

Effect of Dielectric on Capacitance

When a dielectric material is inserted between the plates of a parallel plate capacitor, it significantly affects the capacitance of the capacitor. Let’s understand how the presence of a dielectric modifies the capacitance:

  1. Capacitance in Vacuum (No Dielectric):
    • In the absence of a dielectric (vacuum), the capacitance of a parallel plate capacitor is given by: C₀ = ε₀ * (A / d) where C₀ is the capacitance, ε₀ (epsilon-zero) is the vacuum permittivity constant (approximately 8.85 x 10⁻¹² F/m), A is the area of each plate, and d is the separation between the plates.
    • The vacuum capacitance C₀ depends on the physical characteristics of the capacitor, such as plate area and separation.
  2. Introduction of Dielectric:
    • When a dielectric material is fully inserted between the plates of the capacitor, the dielectric gets polarized by the electric field.
    • This polarization results in the creation of induced surface charges on the dielectric surfaces facing the plates: sp (positive) and -sp (negative). These induced charges are equivalent to adding two charged sheets at the dielectric boundaries.
  3. Change in Electric Field Inside Dielectric:
    • Due to the induced surface charges on the dielectric, the electric field inside the dielectric becomes different from the electric field in vacuum.
    • The electric field inside the dielectric corresponds to a net surface charge density of ±(s – sp) on the plates, where s is the original surface charge density of the plates.
  4. Potential Difference Across Plates with Dielectric:
    • With the dielectric in place, the potential difference (voltage) across the plates of the capacitor is calculated as follows:sV = (ε₀E₀d) - (ε₀Ed) = ε₀Ed(d - sp/ε₀) where E₀ is the electric field without the dielectric, and E is the electric field inside the dielectric.
  5. Change in Capacitance:
    • The capacitance (C) of the capacitor with the dielectric is defined as the ratio of the charge on one of the plates (Q) to the voltage across the plates (V): C = Q / V
    • Substituting the expressions for V and Q, we get C = ε₀A / (d - sp/ε₀)
  6. Dielectric Constant (Permittivity) – ε:
    • The term ε₀A / d represents the vacuum capacitance C₀. Therefore, the capacitance with the dielectric (C) can be expressed as:mathematicaCopy codeC = C₀ / (1 - sp / (ε₀A))
    • We define the dielectric constant (ε) as the ratio of the capacitance with the dielectric (C) to the vacuum capacitance (C₀):mathematicaCopy codeε = C / C₀ = 1 / (1 - sp / (ε₀A))
  7. Significance of Dielectric Constant:
    • The dielectric constant (ε) of a substance is a dimensionless constant that quantifies how much the presence of the dielectric increases the capacitance of the capacitor compared to a vacuum.
    • Dielectric constants are always greater than 1 because the introduction of a dielectric increases the capacitance.
    • The dielectric constant varies for different materials and is a measure of the ability of a material to be polarized by an electric field.

Combination of Capacitors

Capacitors can be combined in various ways to obtain a system with an effective capacitance (C) different from the individual capacitances (C₁, C₂, …, Cₙ). Here are two common ways capacitors can be combined:

1. Capacitors in Series:

When capacitors are connected in series, they share the same charge but have different potential differences. The total potential difference across the combination is the sum of the potential differences across each capacitor.

For two capacitors (C₁ and C₂) in series:

  • Charge on both capacitors is the same (Q₁ = -Q₂).
  • The total potential difference (V) across the combination is V = V₁ + V₂.
  • The effective capacitance (C) of the combination is defined as the ratio of the charge to the potential difference, C = Q / V.

Using these relationships, we can derive the formula for the effective capacitance of capacitors in series:

1/C = 1/C₁ + 1/C₂

This formula can be extended to any number of capacitors in series:

1/C = 1/C₁ + 1/C₂ + ... + 1/Cₙ

2. Capacitors in Parallel:

When capacitors are connected in parallel, they have the same potential difference but may carry different charges. The total charge on the combination is the sum of the charges on each capacitor.

For two capacitors (C₁ and C₂) in parallel:

  • Potential difference across both capacitors is the same (V).
  • The total charge (Q) on the combination is Q = Q₁ + Q₂.
  • The effective capacitance (C) of the combination is defined as the ratio of the charge to the potential difference, C = Q / V.

Using these relationships, we can derive the formula for the effective capacitance of capacitors in parallel:

C = C₁ + C₂

This formula can be extended to any number of capacitors in parallel:

C = C₁ + C₂ + ... + Cₙ

These formulas provide a way to calculate the effective capacitance when capacitors are combined in series or in parallel. When capacitors are combined in series, the effective capacitance is always less than the smallest individual capacitance. When combined in parallel, the effective capacitance is the sum of the individual capacitances. These principles are useful for designing circuits and calculating capacitance values in practical applications.

Energy Stored in a Capacitor

A capacitor stores energy in the form of electric potential energy, which is related to the work done in transferring charge from one conductor to another. To calculate the energy stored in a capacitor, consider the following steps:

  1. Initially, two uncharged conductors (Conductor 1 and Conductor 2) are present. No charge is on either conductor.
  2. Charge is transferred from Conductor 2 to Conductor 1 bit by bit until Conductor 1 accumulates a charge of Q, and Conductor 2 has a charge of -Q.
  3. During this process, external work is done to transfer positive charge from Conductor 2 to Conductor 1. The work done in transferring an infinitesimal amount of charge dQ from Conductor 2 to Conductor 1 is given by: dW=QdQ/C. ​Where:
    • dW is the work done,
    • Q is the charge on Conductor 1 (or -Q on Conductor 2),
    • dQ is the infinitesimal charge being transferred, and
    • C is the capacitance of the system.
  4. Integrating this equation over the entire process of transferring charge from 0 to Q: W=∫​QdQ/C​
  5. Solving the integral gives us the work done or the potential energy stored in the capacitor: U=1/2 * Q^2/C. ​Where U is the potential energy stored in the capacitor.

This formula shows that the potential energy stored in a capacitor depends on the square of the charge (Q) and the reciprocal of the capacitance (C).

Additionally, the energy stored in the capacitor can also be expressed in terms of the electric field (E) between the plates. For a parallel-plate capacitor with area A and separation d, the energy stored is:

U=1/2 * ϵ0​E^2 * Ad

Where:

  • U is the energy stored in the capacitor.
  • ϵ0​ is the permittivity of free space.
  • E is the electric field between the plates.
  • A is the area of each plate.
  • d is the separation between the plates.

This formula highlights the energy density of the electric field (u), which is the energy stored per unit volume of space:

u=1/2 * ϵ0​E^2

This energy density of the electric field is a general result applicable to electric fields created by various charge configurations, not just parallel-plate capacitors. It describes how much energy is stored in the electric field per unit volume.

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