Chapter: Electric Fields and Capacitors
Welcome to Electric Fields and Capacitors! This chapter forms a core part of Unit 4 (IA2): Further Mechanics, Fields and Particles. Here, we move beyond basic circuit rules and investigate what invisible electric fields look like, how charges interact through empty space, how we store electrical energy in devices called capacitors, and how charges discharge over time. Don't worry if the maths looks intimidating at first glance—we will break down every single equation step-by-step.
Note on syllabus placement: This chapter focuses purely on static electric fields, potentials, and capacitor circuits. Magnetic forces and electromagnetic induction are covered in detail in the companion chapter Magnetic Fields and Electromagnetic Induction.
---1. Electric Fields and Electric Field Strength
What is an Electric Field?
An electric field is defined as a region where a charged particle experiences a force. Just as a mass experiences a force in a gravitational field, an electric charge feels an electrostatic push or pull whenever it enters an electric field.
Electric Field Strength (\(E\))
We quantify the strength of an electric field by measuring the force it exerts per unit of charge placed in that field.
By definition: Electric field strength (\(E\)) at a point is the force per unit positive charge acting on a stationary test charge at that point.
\(E = \frac{F}{Q}\)
Where:
• \(E\) = electric field strength (measured in \(\text{N C}^{-1}\) or \(\text{V m}^{-1}\))
• \(F\) = electrostatic force acting on the charge (in newtons, \(\text{N}\))
• \(Q\) = magnitude of the charge experiencing the force (in coulombs, \(\text{C}\))
Important Vector Rule: Force and electric field strength are vectors. The direction of an electric field is defined as the direction of the force that a positive test charge would experience.
• A positive charge accelerates in the same direction as the field lines.
• A negative charge (such as an electron) accelerates in the opposite direction to the field lines.
Key Takeaway
Electric field strength is force per unit charge: \(E = \frac{F}{Q}\). If you know the field strength \(E\), the force on any charge \(Q\) is simply \(F = EQ\).
---2. Coulomb's Law and Radial Fields
Coulomb's Law
When you have two point charges, \(Q_1\) and \(Q_2\), separated by a distance \(r\), they exert an equal and opposite electrostatic force on each other. Coulomb's law states that the electrostatic force between two point charges is directly proportional to the product of their charges and inversely proportional to the square of the distance between their centres.
\(F = \frac{Q_1 Q_2}{4\pi\varepsilon_0 r^2}\)
Where:
• \(F\) = electrostatic force (\(\text{N}\))
• \(Q_1, Q_2\) = magnitudes of the two interacting charges (\(\text{C}\))
• \(r\) = separation between the centres of the charges (\(\text{m}\))
• \(\varepsilon_0\) = permittivity of free space (\(8.85 \times 10^{-12}\text{ F m}^{-1}\))
• \(\frac{1}{4\pi\varepsilon_0}\) = Coulomb law constant (\(8.99 \times 10^9\text{ N m}^2\text{ C}^{-2}\))
Exam Tip: Look at the denominator: \(r^2\). This is an inverse-square law! If you double the distance between two charges (\(2r\)), the force drops to a quarter (\(\frac{1}{4}F\)). If you triple the distance (\(3r\)), the force drops to one-ninth (\(\frac{1}{9}F\)).
Electric Field Strength around a Point Charge
By substituting Coulomb's law (\(F = \frac{Q q}{4\pi\varepsilon_0 r^2}\)) into the definition of field strength (\(E = \frac{F}{q}\)), we obtain the equation for the field of a point charge \(Q\):
\(E = \frac{Q}{4\pi\varepsilon_0 r^2}\)
This creates a radial field:
• For an isolated positive charge: field lines point radially outwards (pushing a positive test charge away).
• For an isolated negative charge: field lines point radially inwards (attracting a positive test charge).
Key Takeaway
Point charges produce radial fields obeying an inverse-square law. Double your distance from a point charge, and the field strength drops by a factor of four.
---3. Uniform Fields and the Relation between Field and Potential
Uniform Electric Field Between Parallel Plates
When two flat metal plates are placed parallel to each other and connected across a potential difference \(V\), they produce a uniform electric field in the gap between them (away from the plate edges).
In a uniform field:
• The electric field strength is constant in magnitude and direction at every point.
• Field lines are straight, parallel, and equally spaced, pointing directly from the positive plate to the negative plate.
The field strength between two parallel plates separated by distance \(d\) is given by:
\(E = \frac{V}{d}\)
Where:
• \(V\) = potential difference between the plates (\(\text{V}\))
• \(d\) = distance separating the plates (\(\text{m}\))
This shows why electric field strength can be measured in both \(\text{N C}^{-1}\) and \(\text{V m}^{-1}\); the two units are completely equivalent (\(1\text{ N C}^{-1} = 1\text{ V m}^{-1}\)).
Relation Between Field Strength and Electric Potential
Electric potential \(V\) is related to electric field strength \(E\). In general, the field strength equals the negative gradient of potential with respect to distance:
\(E = -\frac{\Delta V}{\Delta r}\)
This means electric field lines always point in the direction of the steepest decrease in potential (from high potential to low potential).
Radial Electric Potential
For a point charge, the radial potential \(V\) at a distance \(r\) is given by:
\(V = \frac{Q}{4\pi\varepsilon_0 r}\)
Notice that potential depends on \(\frac{1}{r}\), not \(\frac{1}{r^2}\)! Unlike field strength, electric potential is a scalar quantity (it has magnitude and sign, but no direction):
• If \(Q\) is positive, \(V\) is positive everywhere and approaches zero as \(r \to \infty\).
• If \(Q\) is negative, \(V\) is negative everywhere and approaches zero as \(r \to \infty\).
Field Lines and Equipotential Surfaces
Equipotentials are lines or surfaces connecting points that have the exact same electric potential.
Key properties of equipotentials:
• No work is done (\(W = Q\Delta V = 0\)) when moving a charge along an equipotential surface.
• Electric field lines always cross equipotential lines at right angles (\(90^\circ\)).
• In a uniform field: equipotentials are flat planes parallel to the metal plates, equally spaced.
• In a radial field: equipotentials are concentric spheres centered on the point charge. Because \(V \propto \frac{1}{r}\), equipotentials representing equal steps in potential get progressively further apart as you move away from the charge.
Quick Review: Radial vs Uniform Fields
• Uniform field: \(E = \frac{V}{d}\) (constant everywhere; parallel field lines).
• Radial field: \(E = \frac{Q}{4\pi\varepsilon_0 r^2}\) (decreases as \(\frac{1}{r^2}\); lines spread outward or inward).
• Radial potential: \(V = \frac{Q}{4\pi\varepsilon_0 r}\) (decreases as \(\frac{1}{r}\)).
4. Capacitance and Energy Stored
What is a Capacitor?
A capacitor is an electrical component designed to store electric charge and electrical potential energy. It typically consists of two conducting plates separated by an insulating gap or dielectric material.
Definition of Capacitance
The capacitance (\(C\)) of a capacitor is defined as the charge stored per unit potential difference across it:
\(C = \frac{Q}{V}\)
Where:
• \(C\) = capacitance in farads (\(\text{F}\)) (\(1\text{ F} = 1\text{ C V}^{-1}\))
• \(Q\) = magnitude of charge stored on either plate (\(\text{C}\))
• \(V\) = potential difference across the plates (\(\text{V}\))
Note: A farad is a very large unit. You will routinely encounter microfarads (\(\mu\text{F} = 10^{-6}\text{ F}\)), nanofarads (\(\text{nF} = 10^{-9}\text{ F}\)), and picofarads (\(\text{pF} = 10^{-12}\text{ F}\)).
Derivation of Energy Stored in a Capacitor
When charging a capacitor, work must be done against the electrostatic repulsion of the charges already accumulated on the plates. As charge builds up, the potential difference \(V\) across the plates increases proportionally since \(V = \frac{Q}{C}\).
Consider a graph of potential difference \(V\) (y-axis) against charge \(Q\) (x-axis):
• The graph is a straight line through the origin with gradient \(\frac{1}{C}\).
• Work done to add an infinitesimal charge \(\Delta Q\) at potential \(V\) is \(\Delta W = V \Delta Q\).
• The total electrical energy \(W\) stored is the area under the potential difference–charge graph, which is a triangle:
\(\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}\)
\(W = \frac{1}{2}QV\)
Using the definition \(C = \frac{Q}{V}\), we can substitute for \(Q\) (\(Q = CV\)) or for \(V\) (\(V = \frac{Q}{C}\)) to obtain the three official forms for stored energy:
\(W = \frac{1}{2}QV = \frac{1}{2}CV^2 = \frac{Q^2}{2C}\)
Why isn't the energy simply \(QV\)? A power supply delivers energy \(W_{\text{battery}} = QV\) when shifting charge \(Q\) across potential \(V\). However, the capacitor only stores \(\frac{1}{2}QV\). The remaining 50% of the energy is dissipated as thermal energy in the connecting wires and circuit resistance during charging!
Key Takeaway
Capacitance is charge per unit voltage: \(C = \frac{Q}{V}\). The energy stored equals the triangular area under the \(V\)-\(Q\) graph: \(W = \frac{1}{2}QV = \frac{1}{2}CV^2 = \frac{Q^2}{2C}\).
---5. Resistor-Capacitor (RC) Circuits: Charging and Discharging
Charging a Capacitor through a Resistor
When a capacitor of capacitance \(C\) is connected in series with a resistor \(R\) and a DC power supply of e.m.f. \(V_0\):
1. At time \(t = 0\): The capacitor is uncharged (\(Q = 0\), \(V_C = 0\)). The entire supply voltage drops across the resistor, so the initial charging current is at its maximum: \(I_0 = \frac{V_0}{R}\).
2. As time passes: Charge builds up on the plates. The capacitor p.d. \(V_C\) increases, opposing the supply e.m.f. This reduces the p.d. across the resistor (\(V_R = V_0 - V_C\)), so current \(I\) decreases.
3. As \(t \to \infty\): The capacitor becomes fully charged: \(V_C = V_0\), \(Q = Q_0 = CV_0\), and the current drops to zero (\(I = 0\)).
• The curves for \(Q\) and \(V_C\) against time rise rapidly at first and then level off asymptotically towards maximum values (\(Q_0\) and \(V_0\)).
• The charging current \(I\) against time decays exponentially from \(I_0\) towards zero.
Discharging a Capacitor through a Resistor
When a charged capacitor with initial charge \(Q_0\) and p.d. \(V_0\) is disconnected from the power supply and connected across a resistor \(R\), it discharges through the resistor:
• Charge flows off the plates through the resistor, causing the potential difference and charge to fall simultaneously.
• Because \(I = \frac{V}{R}\) and \(V = \frac{Q}{C}\), the rate of discharge is proportional to the amount of charge remaining: \(\frac{\Delta Q}{\Delta t} \propto -Q\).
• This produces classic exponential decay curves for charge, potential difference, and current!
The exponential discharge equations are:
\(Q = Q_0 e^{-\frac{t}{RC}}\)
\(V = V_0 e^{-\frac{t}{RC}}\)
\(I = I_0 e^{-\frac{t}{RC}}\)
Where:
• \(Q_0, V_0, I_0\) = initial values at \(t = 0\)
• \(Q, V, I\) = values remaining at time \(t\)
• \(e\) = base of natural logarithms (\(\approx 2.718\))
• \(RC\) = the time constant of the circuit
The Significance of the Time Constant (\(\tau = RC\))
The product of resistance and capacitance, \(RC\), has units of time (seconds):
\(\text{Time constant } \tau = RC\)
Unit check: \(\text{Ohms} \times \text{Farads} = \left(\frac{\text{V}}{\text{A}}\right) \times \left(\frac{\text{C}}{\text{V}}\right) = \frac{\text{C}}{\text{A}} = \frac{\text{A s}}{\text{A}} = \text{seconds (s)}\).
What happens when time \(t = RC\)?
\(Q = Q_0 e^{-1} = \frac{Q_0}{e} \approx 0.37 Q_0\)
Definition: The time constant (\(RC\)) is the time taken for the charge, p.d., or current to fall to \(\frac{1}{e}\) (approximately 37%) of its initial value during discharge.
• A larger \(RC\) value means a slower charge and discharge.
• In practical terms, a capacitor is considered virtually fully discharged (less than 1% remaining) after about \(5RC\) (\(e^{-5} \approx 0.0067\)).
Logarithmic Forms and Graphical Analysis
In exam questions, you are frequently asked to linearise exponential discharge equations to find \(RC\) or initial values from experimental data.
Taking the natural logarithm (\(\ln\)) of both sides of \(Q = Q_0 e^{-\frac{t}{RC}}\):
\(\ln Q = \ln\left(Q_0 e^{-\frac{t}{RC}}\right)\)
\(\ln Q = \ln Q_0 - \frac{t}{RC}\)
Rearranging into the standard straight-line form \(y = mx + c\):
\(\ln Q = \left(-\frac{1}{RC}\right)t + \ln Q_0\)
Similarly, for potential difference and current:
\(\ln V = \left(-\frac{1}{RC}\right)t + \ln V_0\)
\(\ln I = \left(-\frac{1}{RC}\right)t + \ln I_0\)
When plotting \(\ln V\) against \(t\):
• The graph is a straight line with a negative gradient.
• Gradient \(= -\frac{1}{RC}\)
• y-intercept \(= \ln V_0\)
• To find the time constant: \(RC = -\frac{1}{\text{gradient}}\).
Core Practical 11: Investigating Capacitor Discharge
In Core Practical 11 (potential difference across a capacitor charging and discharging through a resistor, on an oscilloscope or data logger):
• A circuit is assembled with a DC power supply, a capacitor, a fixed resistor, and a double-throw switch.
• When the switch is thrown to discharge, a data logger or storage oscilloscope records the p.d. \(V\) across the capacitor as a function of time \(t\).
• An oscilloscope or data logger is essential when \(RC\) is small, as regular voltmeters and stopwatches cannot respond quickly enough.
• By calculating \(\ln V\) for each value and plotting \(\ln V\) against \(t\), candidates verify the exponential model and extract an experimental value for the time constant \(RC\) from the gradient.
Key Takeaway
Capacitor discharge is exponential: \(Q = Q_0 e^{-\frac{t}{RC}}\). At \(t = RC\), values drop to \(37\%\) of their starting level. The equation linearises to \(\ln V = \ln V_0 - \frac{t}{RC}\), where the gradient is \(-\frac{1}{RC}\).
---6. Summary of Key Formulas for Revision
Keep this quick-reference table handy when tackling practice questions:
• Electric field strength (general): \(E = \frac{F}{Q}\)
• Coulomb's law force: \(F = \frac{Q_1 Q_2}{4\pi\varepsilon_0 r^2}\)
• Radial electric field strength: \(E = \frac{Q}{4\pi\varepsilon_0 r^2}\)
• Uniform electric field strength: \(E = \frac{V}{d}\)
• Radial electric potential: \(V = \frac{Q}{4\pi\varepsilon_0 r}\)
• Capacitance definition: \(C = \frac{Q}{V}\)
• Energy stored in a capacitor: \(W = \frac{1}{2}QV = \frac{1}{2}CV^2 = \frac{Q^2}{2C}\)
• Exponential discharge: \(Q = Q_0 e^{-\frac{t}{RC}}\), \(V = V_0 e^{-\frac{t}{RC}}\), \(I = I_0 e^{-\frac{t}{RC}}\)
• Logarithmic form: \(\ln Q = \ln Q_0 - \frac{t}{RC}\)