StudentAidUnit 16: Electrostatics — Long Questions
12th Class Physics · Unit 16: Electrostatics
Consider a positive point charge q, with its radial field. Take two points A and B very close to each other, at distances rA and rB from q, with r the distance of the midpoint between them so that the field E can be taken as nearly constant over the small interval. The magnitude of the field at the midpoint is E = (1/4πε₀)(q/r²). Since A and B are very close, taking r as the geometric mean of rA and rB gives r² = rA rB, so E = (1/4πε₀)[q/(rA rB)]. Moving a unit positive charge from B to A, the work done equals the potential difference: VA − VB = −E(rA − rB) = −(1/4πε₀)[q/(rA rB)](rA − rB) = (1/4πε₀)q[(1/rA) − (1/rB)]. To get the absolute potential at A, take B at infinity so that rB → ∞ and VB = 0. This gives the general expression for the electric potential at a distance r from a point charge q: V = (1/4πε₀)(q/r). The potential is positive for a positive charge and negative for a negative charge; for several charges, the net potential at a point is the algebraic sum of the potentials due to each individual charge.
A capacitor is a device that stores electrical energy in the form of an electric field, by storing charge at a certain potential difference; its capacitance C = Q/V measures its ability to store charge per unit potential. A parallel plate capacitor consists of two parallel conducting plates, each of area A, separated by a small distance d, with vacuum/air or a dielectric between them. With vacuum/air as the medium, the uniform field between the plates is E = σ/ε₀, where σ = Q/A is the surface charge density, and V = Ed. Combining these: V = (Q/Aε₀)d, so C = Q/V = Aε₀/d. When a dielectric of relative permittivity (dielectric constant) εᵣ is inserted between the plates, the capacitance increases by the factor εᵣ: C = Aε₀εᵣ/d. The role of the dielectric is to get polarized by the field, producing induced surface charges opposite to the plate charges; this reduces the net field and potential difference for the same charge, thereby increasing the capacitance (C = Q/V) by the factor εᵣ, and also increases the maximum charge the capacitor can safely hold.
Parallel combination Capacitors C₁, C₂, C₃ are connected so that the same potential difference V exists across each (all connected between the same two nodes of the battery). Each carries a different charge: Q₁=C₁V, Q₂=C₂V, Q₃=C₃V. The total charge supplied by the battery is Q = Q₁+Q₂+Q₃, which must equal the charge on the equivalent capacitor, Cp V. So Cp V = C₁V + C₂V + C₃V, giving Cp = C₁ + C₂ + C₃: in parallel, the combined capacitance equals the sum of the individual capacitances.
Series combination Capacitors C₁, C₂, C₃ are connected one after another so each carries the same charge Q (by induction), but the potential difference across each is different, V₁, V₂, V₃, with V = V₁+V₂+V₃ equal to the battery voltage. Since V = Q/C for each, V₁=Q/C₁, V₂=Q/C₂, V₃=Q/C₃, and for the equivalent capacitor V=Q/Cs. Substituting: Q/Cs = Q/C₁ + Q/C₂ + Q/C₃, giving 1/Cs = 1/C₁ + 1/C₂ + 1/C₃: in series, the reciprocal of the equivalent capacitance equals the sum of the reciprocals of the individual capacitances.
Since Q ∝ V for a capacitor, a graph of V versus Q is a straight line through the origin. As the capacitor is charged, each small increment of charge ΔQ is moved against the existing potential difference V at that stage, requiring work ΔW = V·ΔQ, which equals the area of a thin rectangular strip under the V–Q line. The total work done to charge the capacitor from 0 up to a final charge Q (at final potential V) is the sum of all such strips, i.e. the total area under the V–Q graph, which is the area of the right triangle with base Q and height V: Area = ½QV. Therefore the energy stored is W = ½QV. Substituting Q = CV gives W = ½CV², and substituting V = Q/C gives W = Q²/2C. Using V = Ed and C = ε₀εᵣA/d, this can also be written as W = ½ε₀εᵣE²(Ad), showing the energy stored is proportional to the volume (Ad) between the plates, i.e. it resides in the electric field.
[An RC circuit has a battery of voltage V₀, a resistor R, a capacitor C, and a two-way switch S that can connect the circuit either to the battery (position A, for charging) or short the capacitor through R alone (position B, for discharging).] Charging: when the switch is set to A, the battery begins charging the capacitor through R. The charge does not build up instantly; it grows from q=0 at t=0 and approaches the equilibrium value q₀=CV₀ asymptotically. A graph of q versus t is a curve that rises steeply at first and then flattens out as it approaches q₀, the rate of rise being slower for larger RC. Discharging: when the switch is set to B, the capacitor discharges through R; charge flows from the positive plate through the resistor to neutralize the negative plate. A graph of q versus t for discharge starts at q=CV₀ at t=0 and decays exponentially toward zero, following q = q₀e^(−t/RC); the decay is faster for smaller RC. The time constant τ = RC is the time taken for the discharging charge to fall to about 0.37 (37%) of its initial value q₀ (equivalently, for the charging capacitor's charge to rise to about 0.63 of q₀). A smaller time constant means faster charging/discharging; a larger time constant means a slower response.
Electric fans and motors a capacitor connected to the starting winding of a fan or motor charges up when power is switched on and provides an extra boost of power to help the motor overcome static friction and inertia so it starts and spins quickly; the same principle is used in washing machines, blenders, choppers, grinders and water-drawing motors. Refrigerators and air conditioners: their compressors use single-phase motors that struggle to start alone, so a high-capacitance "start capacitor" stores energy and releases it to the compressor motor winding as a power boost to get it rotating; once running, a relay disconnects it, while a "run capacitor" often stays in the circuit to improve efficiency. RC combinations are also used as time-delay circuits to protect such appliances from power fluctuations. Flashguns: a capacitor in a camera flash accumulates charge from the battery and, when the shutter is pressed, rapidly discharges through the flash tube, providing the high current needed to ionize the xenon gas and produce a burst of bright light. Rectification circuits: capacitors are used in bridge/rectifier circuits to help convert alternating current (AC) into direct current (DC). Medical and imaging equipment: capacitors store the energy delivered as a life-saving shock in defibrillators, and help filter noise and improve signal/image clarity in ECG, EEG, and imaging machines (CT, MRI, X-ray, ultrasound).