Class 12 Physics Revision Notes: Electrostatic Potential and Capacitance

Electric Potential

• Electrostatic potential at a point in space is defined as the amount of work done in bringing a unit positive charge from infinity to that point in an electric field, without any acceleration.
• Electrostatic potential is a scalar quantity and is denoted by $V$.
• The formula for electrostatic potential $V$ due to a point charge $q$ at a distance $r$ from the charge is:   $V = \frac{k q}{r}$
  • $V$ = Electrostatic potential
  • $k$ = Coulomb's constant ($9 \times 10^9\,\text{N}\cdot\text{m}^2/\text{C}^2$)
  • $q$ = Source charge
  • $r$ = Distance from the source charge
• The SI unit of electrostatic potential is the volt ($\text{V}$).
• One volt is equivalent to one joule per coulomb ($1\,\text{V} = 1\,\text{J/C}$).
• Definition of 1 Volt: Represents the potential difference when 1 joule of work is done to move 1 coulomb of charge between two points in an electric field.

Electrostatic Potential Difference

• Electrostatic Potential Difference between two points in an electric field is the amount of work done in moving a unit positive charge from one point to another.
• It represents the difference in electrostatic potential between two points and is a scalar quantity.
• Formula for potential difference $\Delta V$ between two points $A$ and $B$ in an electric field:   $\Delta V = V_B - V_A = \frac{W_{AB}}{q}$
  • $\Delta V$ = Potential difference between points $A$ and $B$
  • $V_B$ and $V_A$ = Electrostatic potentials at points $B$ and $A$, respectively
  • $W_{AB}$ = Work done in moving a charge $q$ from $A$ to $B$
  • $q$ = Magnitude of the charge
• Properties of Electrostatic Potential Difference:
  • Scalar Quantity: It does not have direction, only magnitude.
  • Independent of Path: In an electrostatic field, the potential difference between two points depends only on the position of the points and not on the path taken by the charge.
  • Work Done: Measures the work required to move a unit positive charge between two points. A positive potential difference implies work is done against the field, while a negative potential difference implies the field does work.
  • SI Unit: The SI unit of electrostatic potential difference is the volt ($\text{V}$).
  • Relation to Electric Field:     $\Delta V = -\int_A^B \mathbf{E} \cdot d\mathbf{r}$     where $\mathbf{E}$ is the electric field and $d\mathbf{r}$ is the displacement vector along the path from $A$ to B$.\n\n# Electric Potential Due to Point Charge, Dipole, and System of Charges\n\n- Electric Potential Due to a Point Charge:\n - The electric potential at a distance r$from a point charge$q is the work done to bring a unit positive charge from infinity to that point.\n - Formula:\n    V = \frac{k q}{r}\n    where V$is electric potential,$k$is Coulomb's constant ($9 \times 10^9\,\text{N}\cdot\text{m}^2/\text{C}^2$),$q$is charge of the point, and$r is distance from the charge.\n- Electric Potential Due to a Dipole:\n - An electric dipole consists of two equal and opposite charges separated by a small distance d.\n - The potential at a point due to an electric dipole depends on the position relative to the dipole axis.\n - Formula:\n    V = \frac{1}{4\pi\epsilon_0} \frac{p \cos(\theta)}{r^2}\n    where V$is electric potential,$p$is dipole moment ($p = q \times d$with charge$q$and separation$d$),$\theta$is angle between the position vector and the dipole axis,$r$is distance from the center of the dipole, and$\epsilon_0 is permittivity of free space.\n- Electric Potential Due to a System of Charges:\n - For a system of charges, the total electric potential at a point is the algebraic sum of the potentials due to individual charges.\n - Formula:\n    V = \frac{1}{4\pi\epsilon_0} \sum_{i=1}^n \frac{q_i}{r_i}\n    where V$is total electric potential at a point,$q_i$represents individual charges in the system,$r_i$is distance of each charge$q_i$from the point, and$\epsilon_0 is permittivity of free space.\n- Key Summary Points:\n - Point Charge: Potential varies inversely with distance r.\n - Dipole: Potential depends on both distance r$and angle$\theta.\n - System of Charges: The total potential is the algebraic sum of potentials due to individual charges.\n\n# Equipotential Surface\n\n- An equipotential surface is a surface where the electric potential at every point is the same.\n- If a charge is moved along an equipotential surface, no work is done by or against the electric field because the potential difference between any two points on the surface is zero.\n\n\n\n- Key Properties of Equipotential Surfaces:\n - No Work Done: Moving a charge along an equipotential surface requires no work, as potential difference is zero.\n - Perpendicular to Electric Field: Equipotential surfaces are always perpendicular to the electric field lines. Otherwise, work would be done if the charge moved in the direction of the field.\n - Closer Surfaces, Stronger Field: In regions where equipotential surfaces are closer together, the electric field is stronger because potential changes more rapidly over a small distance.\n - Shape Depends on Charge Distribution:\n - For a point charge, equipotential surfaces are concentric spheres around the charge.\n - For a uniform electric field, equipotential surfaces are parallel planes.\n - For a dipole, equipotential surfaces are complex but symmetric.\n\n# Relation Between Field and Potential\n\n- The electric field is the negative gradient of the electric potential.\n- The electric field represents how quickly and in what direction the potential is changing in space.\n- Formula:\n  E = -\frac{dV}{dr}\n - E = Electric field strength\n - dV = Change in electric potential\n - dr = Small displacement in the direction of the field\n\n# Potential Energy in an External Electric Field\n\n- Refers to the energy stored in a charge due to its position in an external electric field, arising from interaction between the charge and the external electric field.\n- Potential Energy of a Point Charge in an External Electric Field:\n - Formula for potential energy U$of charge$q$at a point with potential$V:\n    U = q V\n - For a uniform electric field where V = E d$($E$being field strength,$d being displacement from reference point):\n    U = q E d\n- Potential Energy of an Electric Dipole in an External Electric Field:\n - An electric dipole in a uniform external electric field has potential energy due to interaction between field E$and dipole moment$p:\n    U = -p E \cos(\theta)\n- Key Behavior Points:\n - A positive charge moves from higher potential to lower potential, decreasing potential energy in the direction of the field.\n - A negative charge moves from lower potential to higher potential, increasing potential energy as it moves against the field.\n - For a dipole, potential energy is minimum when aligned with the electric field (\theta = 0^\circ$) and maximum when perpendicular to the field ($\theta = 90^\circ).\n\n# Conductors and Conductors in Electric Fields\n\n- Conductors are materials allowing free flow of electric charges (electrons) through them due to loosely bound outer electrons.\n- Key Properties of Conductors:\n - Free Electrons: Possess conduction electrons not tightly bound to any specific atom, moving freely throughout the material.\n - Electric Field Inside a Conductor: In electrostatic condition, the electric field inside a conductor is zero (E = 0) because free electrons rearrange to cancel any applied field inside.\n - Charge Distribution: Excess charges reside exclusively on the outer surface due to mutual repulsion.\n - Equipotential Surface: The entire conductor acts as an equipotential surface in electrostatic equilibrium (same potential within and on the surface).\n - Electric Field at the Surface: Field just outside is perpendicular to the surface. For irregular surfaces, electric field is stronger where surface curvature is high (sharp edges or points).\n - Electrostatic Shielding: Phenomenon where a conductor shields its interior from external electric fields because E = 0 inside in electrostatic equilibrium.\n- Conductors in Electric Fields:\n - When placed in an external field, free electrons move and rearrange to cancel the interior field, creating induced charge on the surface and resulting in an altered electric field outside.\n\n\n\n# Dielectrics and Polarisation\n\n- Dielectrics are insulating materials that do not conduct electricity but can be polarized in an external electric field (electric field induces dipole moments without allowing free movement of charge).\n- Polarisation in Dielectrics:\n - When placed in an external electric field, charges within molecules get slightly separated.\n - Polarisation (\mathbf{P}) is the alignment of electric dipoles in the material and is defined as the dipole moment per unit volume.\n\n\n\n- Types of Polarisations:\n - Electronic Polarisation: Displacement of electron cloud relative to the nucleus in atoms; present in all dielectrics.\n - Ionic Polarisation: Shift of positive and negative ions in opposite directions in ionic materials.\n - Orientational (Dipolar) Polarisation: Alignment of permanent dipole moments along field direction in polar dielectrics (e.g., water).\n- Polarisation Vector Formula:\n  \mathbf{P} = \frac{\text{Total Dipole Moment}}{\text{Volume}}\n- Effects of Polarisation:\n - Reduced Electric Field: Induced dipoles produce an internal field opposing the external field, reducing overall field strength inside.\n - Dielectric Constant (Relative Permittivity, \epsilon_r$): Quantifies field reduction as ratio of field in vacuum ($E_0$) to reduced field in dielectric ($E):\n    \epsilon_r = \frac{E_0}{E}\n - Capacitance Increase: Increases charge storage capability when placed between capacitor plates:\n    C = \epsilon_r \cdot C_0\n    where C_0 is capacitance without dielectric.\n- Key Properties of Dielectrics:\n - Insulating Material: Non-conducting but polarizable.\n - Dielectric Strength: Maximum electric field a dielectric can withstand without dielectric breakdown (becoming conductive).\n - Induced Charge: Charges induced by polarisation are bound charges appearing on the surface.\n\n# Capacitors and Capacitance\n\n- Capacitors are devices storing electrical energy in an electric field, consisting of two conductive plates separated by a dielectric insulating material.\n\n\n\n- Capacitance:\n - Ability of a capacitor to store charge per unit voltage.\n - Formula:\n    C = \frac{Q}{V}\n- Capacitance of a Parallel Plate Capacitor Formula:\n  C = \frac{\epsilon_0 \epsilon_r A}{d}\n - C = Capacitance\n - \epsilon_0$= Permittivity of free space ($8.85 \times 10^{-12}\,\text{F/m})\n - \epsilon_r = Relative permittivity (dielectric constant)\n - A = Area of one plate\n - d = Distance between plates\n- Energy Stored in a Capacitor:\n  U = \frac{1}{2} C V^2\n  Represents stored energy in the electric field between plates.\n- Factors Affecting Capacitance:\n - Plate Area (A): Increasing area increases capacitance.\n - Plate Distance (d): Decreasing distance increases capacitance.\n - Dielectric Material: Higher dielectric constant increases capacitance.\n- Types of Capacitors:\n - Ceramic Capacitors: Small, non-polarized, ceramic material.\n - Electrolytic Capacitors: Polarized, high capacitance values, used in power supplies.\n - Tantalum Capacitors: Polarized, higher stability and reliability.\n- Applications of Capacitors:\n - Energy Storage: Flashlights and camera flashes.\n - Filtering: Noise reduction in power supplies and signal processing.\n - Timing Circuits: Combined with resistors to create time delays.\n - Coupling and Decoupling: Passes AC signals while blocking DC signals.\n\n# Parallel Plate Capacitor\n\n- Consists of two large, flat conductive plates placed parallel and separated by small distance d. Applying voltage charges one plate positively and the other negatively, creating a uniform electric field.\n\n\n\n- Capacitance Formula:\n  C = \frac{\epsilon_0 \epsilon_r A}{d}\n - C$= Capacitance in farads ($\text{F})\n - \epsilon_0$= Permittivity of free space ($8.85 \times 10^{-12}\,\text{F/m})\n - \epsilon_r = Relative permittivity (dielectric constant)\n - A$= Area of one plate in square meters ($\text{m}^2)\n - d$= Distance between plates in meters ($\text{m})\n- Key Concepts:\n - Uniform Electric Field Between Plates:\n    E = \frac{V}{d}\n - Charge Stored:\n    Q = C V\n - Energy Stored:\n    U = \frac{1}{2} C V^2\n\n\n\n- Effect of Dielectric:\n - Dielectric material reduces electric field strength, enabling more charge storage for same voltage.\n - Increases capacitance by factor \epsilon_r.\n - \epsilon_r = 1$for vacuum;$\epsilon_r > 1 for all materials.\n\n# Combination of Capacitors\n\n- Capacitors in Series:\n - Connected end-to-end; total equivalent capacitance is reduced and always less than the smallest individual capacitance.\n\n\n\n - Formula:\n    \frac{1}{C_{\text{total}}} = \frac{1}{C_1} + \frac{1}{C_2} + \dots + \frac{1}{C_n}\n - Charge: Same charge on each capacitor (Q_{\text{total}} = Q_1 = Q_2 = \dots = Q_n).\n - Voltage: Total voltage is the sum of individual voltages (V_{\text{total}} = V_1 + V_2 + \dots + V_n).\n\n- Capacitors in Parallel:\n - Connected across same two points; total equivalent capacitance increases and equals the sum of individual capacitances.\n\n\n\n - Formula:\n    C_{\text{total}} = C_1 + C_2 + \dots + C_n\n - Voltage: Same voltage across each capacitor (V_{\text{total}} = V_1 = V_2 = \dots = V_n).\n - Charge: Total charge is sum of individual charges (Q_{\text{total}} = Q_1 + Q_2 + \dots + Q_n).\n\n- Comparison Summary:\n - Series Combination:\n - Capacitance: Decreases (\frac{1}{C_{\text{total}}} = \sum \frac{1}{C_i})\n - Charge: Same charge on each capacitor (Q_1 = Q_2 = \dots = Q_n)\n - Voltage: Sum of individual voltages (V_{\text{total}} = \sum V_i)\n - Primary Application: Increasing voltage handling capability.\n - Parallel Combination:\n - Capacitance: Increases (C_{\text{total}} = \sum C_i)\n - Charge: Charge divided across capacitors (Q_{\text{total}} = \sum Q_i)\n - Voltage: Same voltage across each capacitor (V_1 = V_2 = \dots = V_n)\n - Primary Application: Increasing total charge storage capacity.\n\n- Mixed (Series-Parallel) Combination Procedures:\n - Step 1: Reduce parallel groups to single equivalent capacitances using parallel formula.\n - Step 2: Reduce resulting series groups using series formula.\n - Step 3: Repeat process iteratively until full circuit simplifies to single equivalent capacitance.\n\n# Energy Stored in a Capacitor\n\n- A capacitor stores energy in the electric field between plates when charged, which is released upon discharge.\n- Primary Formula:\n  U = \frac{1}{2} C V^2\n - U$= Energy stored in joules ($\text{J})\n - C$= Capacitance in farads ($\text{F})\n - V$= Voltage in volts ($\text{V})\n- Alternative Formula Forms:\n - Using charge Q$and capacitance$C:\n    U = \frac{Q^2}{2 C}\n - Using charge Q$and voltage$V:\n    U = \frac{1}{2} Q V\n- Key Concepts & Applications:\n - Energy Density (u): Energy stored per unit volume in the electric field:\n    u = \frac{1}{2} \epsilon_0 \epsilon_r E^2\n    where \epsilon_0 \epsilon_r$is total permittivity and$E$$ is electric field strength.
  • Discharging Behavior: Stored energy releases into circuit and potential difference decreases over time.
  • Practical Example: Camera flash charges capacitor gradually and releases energy rapidly to generate a bright burst of light.