Cambridge International AS & A Level Physics: Electric Fields and Capacitance Study Guide

Electric Force Between Point Charges

• The Point Charge Assumption for Spheres: For any point located outside a spherical conductor, the electric charge distributed on the surface or throughout the sphere may be treated as a point charge situated at the sphere's geometric centre. This simplification allows for the application of Coulomb's Law to macroscopic spherical objects, provided the observation point is at a distance $r$ greater than the radius of the sphere.

• Coulomb's Law: This law dictates the magnitude of the electrostatic force between two point charges in free space. It is expressed by the formula:   \n  F = \frac{Q_1 Q_2}{4 \times \beta \times \text{vacuum permittivity} \times r^2}\n     In standard symbolic form as presented in the syllabus:   \n  F = \frac{Q_1 Q_2}{4 \times \text{π} \times \text{ε}_0 \times r^2}\n  

  • Force ($F$): Measured in Newtons ($N$).
  • Charges ($Q_1, Q_2$): Measured in Coulombs ($C$).
  • Permittivity of Free Space ($\text{ε}_0$): A constant representing the capability of a vacuum to permit electric field lines.
  • Separation ($r$): The distance between the centres of the two point charges, measured in metres ($m$).

Electric Field of a Point Charge

• Electric Field Strength ($E$): This is defined as the force per unit positive charge acting on a stationary point charge. For a single point charge in free space, the field strength at a distance $r$ is calculated using:   \n  E = \frac{Q}{4 \times \text{π} \times \text{ε}_0 \times r^2}\n  
  • The field strength follows an inverse square law, meaning the strength of the field decreases rapidly as the distance from the charge increases.
  • The unit for electric field strength is Volts per metre ($V \times m^{-1}$) or Newtons per Coulomb ($N \times C^{-1}$).

Electric Potential and Potential Energy

• Definition of Electric Potential: The electric potential at a specific point in an electric field is defined as the work done per unit positive charge in bringing a small test charge from infinity to that point.

  • Small Test Charge: The test charge must be small so that its own electric field does not significantly perturb or redistribute the charges creating the field being measured.
  • Infinity: This is used as the reference zero point for potential ($V = 0 \text{ at } r = \text{∞}$).

• Electric Field as a Potential Gradient: The electric field strength at a point is numerically equal to the negative of the potential gradient at that point. This relationship is expressed as:   \n  E = - \frac{\text{d}V}{\text{d}r}\n  

  • This indicates that the electric field vector points in the direction of the steepest decrease in electric potential.

• Potential due to a Point Charge: The formula for the electric potential at a distance $r$ from a point charge $Q$ in free space is:   \n  V = \frac{Q}{4 \times \text{π} \times \text{ε}_0 \times r}\n  

  • Unlike the electric field, potential is a scalar quantity and follows an inverse relationship ($1/r$) rather than an inverse square law ($1/r^2$).

• Electric Potential Energy ($E_p$): The concept of electric potential leads to the potential energy shared between two point charges ($Q$ and $q$) separated by a distance $r$. The potential energy is calculated as:   \n  E_p = \frac{Q \times q}{4 \times \text{π} \times \text{ε}_0 \times r}\n  

  • If the charges are of the same sign, the potential energy is positive (work must be done to bring them together). If they are of opposite signs, the potential energy is negative (work is done by the field).

Capacitance and Capacitor Fundamentals

• Definition of Capacitance: Capacitance ($C$) is the ratio of the charge ($Q$) stored on one of the plates of a capacitor (or on a conductor) to the potential difference ($V$) across it. This applies to:

  • Isolated Spherical Conductors: Where the sphere stores charge relative to infinity.
  • Parallel Plate Capacitors: Consisting of two conducting plates separated by an insulator (dielectric).

• Fundamental Formula: The charge, capacitance, and voltage are related by:   \n  C = \frac{Q}{V}\n  

  • Unit: The Farad ($F$), where $1 \text{ Farad} = 1 \text{ Coulomb per Volt}$.

• Capacitors in Series: For capacitors connected in a series arrangement, the total potential difference is the sum of individual potential differences, while the charge on each capacitor is identical ($Q_{total} = Q_1 = Q_2$…). The combined capacitance ($C_{total}$) is derived as:   \n  \frac{1}{C_{total}} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} + \text{…}\n  

• Capacitors in Parallel: For capacitors connected in a parallel arrangement, the potential difference across each capacitor is the same ($V_{total} = V_1 = V_2$…), while the total charge is the sum of the charges on each capacitor. The combined capacitance ($C_{total}$) is derived as:   \n  C_{total} = C_1 + C_2 + C_3 + \text{…}\n  

Energy Stored in a Capacitor

• Energy and Work Done: Charging a capacitor involves moving charge against an increasing potential difference. The work done in this process is stored as electric potential energy in the electric field between the plates.

• Graphical Determination: The electric potential energy ($W$) stored in a capacitor can be determined by calculating the area under a potential-charge ($V$ versus $Q$) graph. Since $V$ is proportional to $Q$, the graph is a straight line through the origin, and the area (a triangle) represents the work done.

• Energy Formulae: The total work done/energy stored is given by the following equivalent equations:   \n  W = \frac{1}{2} \times Q \times V\n     \n  W = \frac{1}{2} \times C \times V^2\n  

  • Substituting $Q = C \times V$ into the first formula yields the second.

Discharging a Capacitor

• Analysis of Discharge Graphs: When a capacitor discharges through a resistor ($R$), the potential difference ($V$), charge ($Q$), and current ($I$) all decrease over time according to an exponential decay pattern.

  • Potential Difference ($V$): Decreases exponentially from the initial voltage $V_0$.
  • Charge ($Q$): Decreases as charge flows off the plates.
  • Current ($I$): Decreases because the driving potential difference ($V$) drops according to Ohm's Law ($I = V / R$).

• The Time Constant ($τ$): The time constant for a capacitor-resistor (RC) circuit is defined as the product of the resistance and the capacitance:   \n  τ = R \times C\n  

  • The time constant represents the time taken for the charge, potential difference, or current to fall to approximately $37$ ($1/e$) of its initial value.

• Exponential Decay Equations: The variation of any quantity $x$ (where $x$ can represent current $I$, charge $Q$, or potential difference $V$) during the discharge process is modelled by:   \n  x = x_0 \times e^{- \frac{t}{R \times C}}\n  

  • $x_0$: The initial value of the quantity at $t = 0$.
  • $t$: The elapsed time since the discharge began.
  • $e$: The base of the natural logarithm (approx. 2.718).