Unit 16: Electrostatics — Short Questions

Nothing can be concluded about the electric intensity from the value of the potential alone — the field is related to the rate of change (gradient) of potential, E = −ΔV/Δr, not to the potential's value itself. So a point where V = 0 need not have E = 0. For example, midway between two equal and opposite point charges the potential is zero, but the electric field there is not zero.

The electron-volt (eV) is a unit of energy, not of potential difference. It is defined as the energy gained by an electron when it is accelerated through a potential difference of 1 volt, so 1 eV = (1.6 × 10⁻¹⁹ C)(1 V) = 1.6 × 10⁻¹⁹ J.

Capacitance is the amount of charge on one plate (or on a conductor) necessary to raise its potential by one volt with respect to the other; C = Q/V. Its SI unit is the farad (F), equal to one coulomb per volt. A capacitor has a capacitance of 1 farad if a charge of 1 coulomb given to one plate produces a potential difference of 1 volt between the plates.

The dielectric constant (relative permittivity, εᵣ) of a material is the ratio of the capacitance of a parallel plate capacitor with that insulating material as the medium between the plates to its capacitance with vacuum (or air) as the medium: εᵣ = C(medium)/C(vacuum).

Since C = Aε₀/d (or Aε₀εᵣ/d with a dielectric), capacitance is inversely proportional to the plate separation d. Increasing the separation between the plates therefore decreases the capacitance.

For the same applied voltage V, the equivalent capacitance in parallel (Cp = C₁+C₂+C₃) is always larger than the equivalent capacitance in series (1/Cs = 1/C₁+1/C₂+1/C₃, which is smaller than the smallest individual capacitance). Since the energy stored is W = ½C_eqV², a larger equivalent capacitance stores more energy. Hence (ii), the parallel combination, stores the greater total energy.

The time constant (τ) of an RC circuit is the product of resistance and capacitance, τ = RC. It is defined as the time required for the charge on a discharging capacitor to fall to about 0.37 (37%) of its initial value Q₀ (equivalently, the time for a charging capacitor's charge to rise to about 0.63 of its equilibrium value).

Electric potential (V) is the work done per unit positive charge in bringing a test charge from infinity to a point; it is a property of the field/source charge at that point alone (unit: volt, J/C). Electric potential energy (U) is the actual energy possessed by a specific charge q placed at that point, U = qV (unit: joule). For example, a point at distance r from a charge Q has a fixed potential V regardless of what is placed there; but if we bring a charge q = 2 μC to that point, it acquires potential energy U = qV, which depends on the value of q as well as V.

A charge free to move in an electric field moves so as to lower its potential energy. Since U = qV, a positive charge accelerates from a point of higher potential to a point of lower potential, while a negative charge moves from lower to higher potential. The decrease in potential energy, ΔU = qΔV, appears as the kinetic energy gained by the charge. Thus, knowing the potential at different points lets us predict the direction a free charge will move without directly computing field vectors.

The negative sign shows that the electric field points in the direction in which potential decreases, i.e. from a region of higher potential to a region of lower potential. For example, between two oppositely charged parallel plates, potential decreases steadily from the positive plate to the negative plate, and the field E = −ΔV/Δd points from the positive plate toward the negative plate, consistent with the negative gradient.

The dielectric (or vacuum/air) gap between the plates is an insulator, so charge genuinely cannot cross it. However, as the battery pushes charge onto one plate, an equal amount of charge is simultaneously drawn away from the other plate (by induction/repulsion), so charge continuously flows through the connecting wires on both sides of the capacitor. This continuous flow of charge in the circuit wires is what is measured as the charging current, even though no charge physically crosses the gap between the plates.

For the same supply voltage V, the parallel combination has equivalent capacitance Cp = C₁+C₂, while the series combination has a smaller equivalent capacitance given by 1/Cs = 1/C₁+1/C₂ (always less than the smaller of C₁, C₂). Since the stored energy is W = ½C_eqV², and Cp > Cs for the same V, the parallel combination stores more total energy than the series combination when connected to the same potential difference.

The energy of a charged capacitor is best understood as being stored in the electric field that exists in the space between the plates, rather than literally on the plates or in the dielectric material. As the capacitor is charged, each small increment of charge ΔQ is moved against the existing potential difference V, requiring work ΔW = V·ΔQ; the sum of this work (the area under the V–Q graph) is W = ½QV = ½CV², which is stored as electric potential energy in the field. Equivalently, the energy can be written as W = ½ε₀εᵣE²(Ad), proportional to the volume occupied by the field, showing it resides in the field itself.

When a dielectric is placed between the plates, the electric field polarizes its atoms, displacing bound electrons slightly relative to their nuclei. This produces induced surface charges on the dielectric that are opposite in sign to the charges on the adjacent plates, which partially cancels the field between the plates. For the same charge Q on the plates, the electric field and hence the potential difference V decrease. Since C = Q/V, a decrease in V with Q unchanged means C increases — by the factor εᵣ, the dielectric constant.