Source: Chapters 7 and 8
Tags: electric potential, voltage, potential energy, potential difference, equipotential, capacitor, capacitance, dielectric, energy storage, series capacitors, parallel capacitors
Difficulty: Intermediate
Prerequisites: Chapters 5-6 notes (electric field, Coulomb's law, Gauss's law). Familiarity with work-energy theorem.
Electric potential assigns a single number (a scalar) to every point in space, instead of a vector. This makes many calculations far simpler than working with the electric field directly. Capacitance then takes these ideas and puts them to work: a capacitor is a device that stores energy in the electric field between its plates. These two chapters are the direct lead-in to circuits, where potential difference (voltage) is the quantity you measure with a voltmeter and capacitors appear in nearly every electronic device.
Electric potential is energy per unit charge. The potential difference between two points tells you how much work the electric field does (or you must do) to move a charge between them. Capacitors store charge and energy by maintaining a potential difference between two conductors, and their behaviour depends on geometry, connection (series vs parallel), and whether a dielectric is present.
Electric potential energy (U)
The energy a charge has because of its position in an electric field. Change in potential energy: ΔU = -W, where W is the work done by the electric force. Think of it as: the electrical equivalent of gravitational potential energy.
Electric potential (V)
Potential energy per unit charge: V = U/q. Measured in volts (1 V = 1 J/C). In simple terms, voltage is how much "push" a charge gets at a given location.
Potential difference (ΔV)
The difference in electric potential between two points: V_B - V_A = -∫ E · dl. This is what a voltmeter measures.
Equipotential surface
A surface on which every point has the same electric potential. No work is done moving a charge along an equipotential. Equipotential surfaces are always perpendicular to electric field lines.
Capacitor
A device consisting of two conductors (plates) separated by a small gap, used to store charge and electrical energy.
Capacitance (C)
The ratio of charge stored to the potential difference across the plates: C = Q/V. Measured in farads (F). Determined entirely by geometry and the material between the plates. Think of it as: how much charge the device can hold per volt applied.
Dielectric
An insulating material placed between capacitor plates. It increases the capacitance by a factor κ (the dielectric constant) and reduces the electric field inside for a given charge. In simple terms, a dielectric lets the capacitor store more charge at the same voltage.
Dielectric constant (κ)
A dimensionless number (always ≥ 1) that describes how much a dielectric material reduces the electric field. C = κC₀, where C₀ is the capacitance without the dielectric.
Energy density (u_E)
Energy per unit volume stored in an electric field: u_E = ½ ε₀E².
Work done by an applied force: W = F · d. The change in potential energy is ΔU = -W (work done by the electric force).
Because the electric force is conservative, only the endpoints matter, not the path.
Relationship to kinetic energy via conservation of energy: K_i + U_i = K_f + U_f.
Potential energy of a dipole in a field: U = -pE cos θ.
V = U/q. The potential at a point due to a point charge Q: V = kQ/r.
Potential difference: V_B - V_A = -∫(A to B) E · dl.
For a uniform field: ΔV = -Ed, where d is the distance along the field direction.
For a collection of point charges, the total potential is the algebraic sum (not a vector sum) of the individual potentials. This is one of the great advantages of working with potential rather than the field.
For continuous charge distributions: V = ∫ k dq / r.
The electric field is the negative gradient of the potential: E = -dV/ds, where s is the direction of steepest change.
In component form: E_x = -∂V/∂x, E_y = -∂V/∂y, E_z = -∂V/∂z.
The field points from high potential to low potential.
Equipotential surfaces are perpendicular to field lines everywhere.
No work is required to move a charge along an equipotential (ΔV = 0).
The entire surface (and interior) of a conductor in equilibrium is an equipotential.
The potential is higher near positive charges and lower near negative charges.
When two conducting spheres of different radii are connected by a wire, charge flows until they reach the same potential.
The smaller sphere ends up with a higher surface charge density and a stronger field at its surface.
A parallel-plate capacitor: C = ε₀A/d, where A is the plate area and d is the separation.
The electric field between the plates: E = σ/ε₀ = V/d.
Capacitance depends only on geometry and the dielectric material, not on the charge or voltage applied.
Parallel: C_total = C₁ + C₂ + C₃ + ... Each capacitor has the same voltage across it (equal to the source voltage). Charge distributes among them.
Series: 1/C_total = 1/C₁ + 1/C₂ + 1/C₃ + ... Each capacitor holds the same charge. The voltage drops add up to the source voltage.
For complex networks, simplify step by step using these rules.
U_C = ½QV = ½CV² = ½Q²/C.
The work done to charge a capacitor equals the energy stored: you move charge from one plate to the other against an increasing potential difference.
Energy density in the electric field: u_E = ½ε₀E².
Inserting a dielectric increases capacitance: C = κC₀.
Energy stored with a dielectric: U = ½CV² (use the new C).
The dielectric reduces the electric field between the plates for a given charge, because induced surface charges inside the dielectric partially cancel the applied field.
The induced field inside the dielectric: E_induced = E₀(1 - 1/κ).
Dielectric breakdown occurs when the external field is so strong that the insulator's molecules ionise and it becomes a conductor. The dielectric strength is the maximum field the material can withstand.
Maximum charge a capacitor can store: Q_max = κC₀V (limited by dielectric breakdown voltage).
Polar molecules have permanent electric dipole moments. In an external field, they align (partially) with the field.
Non-polar molecules develop an induced dipole moment only in the presence of an external field.
The aligned dipoles create induced surface charges on the dielectric surfaces, which produce a field opposing the applied field.
Quantity | Formula |
|---|---|
Potential (point charge) | V = kQ/r |
Potential difference (uniform field) | ΔV = -Ed |
Parallel-plate capacitance | C = ε₀A/d |
Capacitance with dielectric | C = κε₀A/d |
Capacitors in parallel | C_total = C₁ + C₂ + ... |
Capacitors in series | 1/C_total = 1/C₁ + 1/C₂ + ... |
Energy in capacitor | U = ½CV² = ½QV = ½Q²/C |
Energy density | u_E = ½ε₀E² |
Field from potential | E = -dV/ds |
Capacitors are everywhere in electronics: they smooth out voltage fluctuations in power supplies, store energy for camera flashes, and filter signals in audio equipment. The concept of dielectric breakdown is why high-voltage equipment needs careful insulation design, and it is also the principle behind spark plugs in petrol engines.
Students often confuse potential (V, a scalar at a point) with potential energy (U, energy of a charge at that point). V = U/q. A point in empty space has a potential; a charge at that point has potential energy.
A common error with series capacitors is adding the capacitances directly. In series, you add the reciprocals.
Students sometimes think inserting a dielectric always increases the stored energy. If the capacitor is disconnected from the battery (constant Q), inserting a dielectric decreases the energy. If it stays connected (constant V), the energy increases because the capacitance increases at the same voltage.
Equipotential lines are not the same as field lines. They are perpendicular to each other.
⚠️ Know the three equivalent forms of capacitor energy: ½QV, ½CV², ½Q²/C. The exam often gives you two of the three variables (Q, C, V) and asks for energy.
⚠️ Series vs parallel capacitor rules are the opposite of resistor rules. Do not mix them up.
⚠️ Be ready to find the electric field from a given potential function using differentiation (the gradient).
⚠️ Dielectric problems: always check whether the capacitor is connected to a battery (constant V) or isolated (constant Q) before deciding what changes and what stays the same.
True or false: The electric potential at a point midway between two equal positive charges is zero.
False. Both charges contribute a positive potential, so the total is positive. (The electric field is zero there, but the potential is not.)
Fill in the blank: Equipotential surfaces are always ________ to electric field lines.
Perpendicular.
True or false: When capacitors are connected in series, the total capacitance is always less than the smallest individual capacitance.
True.
Fill in the blank: The energy density in an electric field is u_E = ½________.
ε₀E².
Q: Two capacitors, 3 μF and 6 μF, are connected in series to a 12 V battery. What is the charge on each capacitor?
A: 1/C_total = 1/3 + 1/6 = 1/2, so C_total = 2 μF. Q = CV = (2 x 10⁻⁶)(12) = 24 μC on each (same charge in series).
Q: A parallel-plate capacitor with C = 50 pF is charged to 200 V and then disconnected. A dielectric with κ = 4 is inserted. What is the new voltage?
A: Q is constant (disconnected). New C = 4 × 50 = 200 pF. V_new = Q/C_new = (50 × 10⁻¹² × 200)/(200 × 10⁻¹²) = 50 V.
Q: The electric potential in a region is V = 3x² - 2y (in SI units). What is the electric field at the point (1, 2)?
A: E_x = -∂V/∂x = -6x = -6 N/C. E_y = -∂V/∂y = 2 N/C. E = (-6î + 2ĵ) N/C at (1, 2).
Electric potential is the foundation of voltage in circuits (Ch. 9-10). The concept of energy stored in a capacitor leads directly to energy in inductors (Ch. 14) and LC oscillations. Equipotential ideas return in the context of conductors and circuit analysis, where wires are treated as equipotentials.
electric potential, voltage, potential difference, volt, potential energy, work-energy, equipotential, gradient, capacitor, capacitance, farad, parallel plate, dielectric, dielectric constant, dielectric breakdown, energy density, series capacitors, parallel capacitors, charge storage, PHYS 212