BPSC 70th CCE Re-exam Prelims General Studies · Question 55 of 142

Correct — A, Inversely proportional to frequency. Capacitive reactance is defined by X_C = 1 / (ωC) = 1 / (2πfC), where f is the frequency of the supply in hertz and C the capacitance in farads. The frequency sits in the denominator, so doubling the frequency halves the reactance: the relationship is inverse, and the exponent on f is exactly minus one. The physics behind the algebra is worth holding on to, because it is what the formula is describing. A capacitor does not conduct through its dielectric at all; current 'flows' in the circuit only while the plates are charging or discharging. At a high supply frequency the voltage reverses so often that the plates never approach full charge, charge keeps sloshing in and out, and a large alternating current is sustained — which is another way of saying the opposition is small. At a low frequency the plates have time to charge fully, the current falls away as they do, and the opposition is large. Push that to the limit: at f = 0, which is direct current, X_C becomes infinite and the capacitor blocks the circuit completely. That single limit is the memory hook, because it is a fact everybody already knows — a capacitor blocks DC and passes AC — and it can only be true if reactance rises as frequency falls. Since option (a) states the relation exactly, the escape option (d) has nothing to do.

• (b)Directly proportional to frequency — This is the inductor's behaviour, not the capacitor's. Inductive reactance is X_L = ωL = 2πfL, with frequency in the numerator, so an inductor opposes high frequencies and passes low ones — exactly opposite to a capacitor. The two are deliberately mirrored, and swapping them is the standard error this option is built to catch.
• (c)Inversely proportional to sq. root of frequency — A square-root dependence appears nowhere in the reactance formulae. It is imported from the resonance condition of an LC circuit, where f = 1 / (2π√(LC)) puts a square root on the product of inductance and capacitance — a different quantity in a different equation. In X_C = 1/(2πfC) the power of f is one.
• (d)None of the above — This escape is correct only if all three named statements fail, and the first one is the textbook definition. The temptation is to read 'reactance' as 'impedance' and object that the total opposition of an RC circuit, Z = √(R² + X_C²), is not a clean inverse function of frequency — true, but the stem asks for reactance, not impedance.

In a direct-current circuit the only opposition to current is resistance, R, measured in ohms and set by the material and dimensions of the conductor. In an alternating-current circuit two further kinds of opposition appear, and both come from energy being stored and given back rather than dissipated as heat. A capacitor stores energy in the electric field between its plates and opposes a *change* of voltage; an inductor stores energy in the magnetic field around its coil and opposes a *change* of current. The opposition each offers is called reactance, is also measured in ohms, and unlike resistance it depends on the frequency of the supply: X_C = 1/(2πfC) for the capacitor and X_L = 2πfL for the inductor. Combined with resistance they give the impedance Z = √(R² + (X_L − X_C)²), the total opposition of an AC circuit. When X_L equals X_C the two cancel, impedance falls to R alone and the circuit is at resonance — the principle every radio tuner works on.

You do not need to recall the formula to answer this. Recall instead the one behaviour of a capacitor everyone learns first: it blocks direct current and lets alternating current through. Direct current is frequency zero. If a capacitor offers infinite opposition at zero frequency and less opposition as frequency rises, then reactance must fall as frequency rises — an inverse relationship, option (a). Run the same test on an inductor and it comes out the other way: a coil passes DC freely and chokes high frequencies, which is why a choke coil is called a choke. That pair of behaviours is enough to separate options (a) and (b) without algebra. Option (c) then falls because no standard AC quantity carries a square root of frequency alone. This is also where the practical uses come from: a capacitor in series with a signal makes a high-pass filter, an inductor in series makes a low-pass filter, and both are everyday electronics rather than exam abstractions.

• Capacitive reactance X_C = 1/(ωC) = 1/(2πfC), measured in ohms; inductive reactance X_L = ωL = 2πfL, also in ohms.
• At f = 0 (direct current) X_C is infinite, so a capacitor blocks DC; as f rises X_C falls, so a capacitor passes high-frequency AC.
• Impedance of a series RLC circuit is Z = √(R² + (X_L − X_C)²); at resonance X_L = X_C, the reactances cancel and Z = R.
• The resonant frequency of an LC circuit is f = 1/(2π√(LC)) — the only place a square root of a circuit quantity appears, and the source of the trap in option (c).
• Reactance stores and returns energy rather than dissipating it: average power consumed in a pure capacitor or pure inductor over a full AC cycle is zero, because voltage and current are 90 degrees out of phase.

• Swapping the capacitor and the inductor. Frequency is in the denominator for X_C and the numerator for X_L; every other error on this topic follows from getting that round the wrong way.
• Importing the square root from the resonance formula f = 1/(2π√(LC)) into the reactance formula, where the power of f is one.
• Answering for impedance instead of reactance. Impedance combines resistance and reactance under a square root; the stem asks only for the reactive part.

BPSC asks physics as a one-line relation with no numbers — name the quantity, state how it varies, or pick the correct formula, as the 70th CCE paper of December 2024 did with the resistance of a wire. UPSC almost never asks a bare circuit formula; when electricity appears in its prelims it is wrapped in a technology or energy context, such as how a photovoltaic array's direct current is inverted for the grid.

• practice — not a real PYQ

  The reactance of an inductor in an AC circuit is

  • (a)Inversely proportional to frequency
  • (b)Independent of frequency
  • (c)Directly proportional to frequency
  • (d)Inversely proportional to the square of frequency

  Answer(c) Directly proportional to frequency — X_L = 2πfL, the exact mirror of the capacitor's X_C = 1/(2πfC), which is why an inductor is used as a choke against high frequencies.

• practice — not a real PYQ

  A capacitor connected in a circuit carrying direct current behaves as

  • (a)A short circuit, because its reactance is zero
  • (b)An open circuit, because its reactance is infinite
  • (c)A pure resistance equal to 1/C
  • (d)A source of electromotive force

  Answer(b) An open circuit, because its reactance is infinite — at f = 0 the expression 1/(2πfC) diverges, which is exactly why a capacitor blocks DC while passing AC.