Capacitance and Dielectrics: questions with solutions
3 free questions from this chapter (the full set has 30). Marking: Multi-correct: +4 all correct, +1 per correct option if none wrong, −2 if any wrong · Numerical: +4 / 0 · Matching: +3 / −1.
Question 1Level 2 · ExamSingle correct
An air-filled parallel-plate capacitor has plate area A and plate separation d; write C0=ε0A/d. A dielectric slab of dielectric constant K=4, thickness d/2 and area A lies parallel to the plates inside the gap. The capacitor is charged by connecting it to a battery of emf V0, after which the battery is disconnected, leaving the capacitor isolated. The slab is then pulled out slowly (quasi-statically, with no kinetic energy given to it). Edge effects are negligible. The ratio of the work done by the external agent in removing the slab to the energy stored in the capacitor just before the removal is ______. Exactly one option is correct.
A2/5
B3/5
C3/8
D8/5
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Answer: B
With the slab: C1=d−t+t/Kε0A=d/2+d/8ε0A=58C0, so Q=58C0V0 and U1=2C1Q2=54C0V02.
Isolated, so Q stays fixed while the slab is removed: U2=2C0Q2=2532C0V02.
Quasi-static removal with no heat: Wext=U2−U1=2512C0V02, and Wext/U1=4/512/25=53.
Check: in general W/U1=C1/C0−1=8/5−1=3/5; it is positive because the slab is attracted into the gap.
Trap: treating V as fixed (battery still connected) would give C0/C1−1<0 for the change in stored energy.
Difficulty: Hard
Question 2Level 3 · AdvancedSingle correct
Three identical large thin conducting plates P1, P2, P3, each of area A, are held parallel to one another in that order. The gap between P1 and P2 is d and the gap between P2 and P3 is 2d, with d≪A; edge effects are negligible and the plates are far from all other objects. P1 and P3 are initially uncharged and are joined to each other by a thin conducting wire that runs outside the plates (its own capacitance is negligible). P2 is insulated and carries charge +Q. In electrostatic equilibrium, the net charge on plate P3 is ______. Exactly one option is correct.
A−Q/3
B−Q/6
C+Q/6
Dzero
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Answer: C
Let the faces of P2 toward P1 and toward P3 carry qa and qb, with qa+qb=Q. The fields in the gaps are E1=qa/(ε0A) and E2=qb/(ε0A), both pointing away from P2.
P1 and P3 are at the same potential, so E1d=E2(2d), giving qa=2Q/3, qb=Q/3. The facing (inner) faces of P1 and P3 carry −2Q/3 and −Q/3.
The total charge of the whole system is Q, so each outermost face (outer face of P1 and outer face of P3) carries Q/2 (the field outside the stack is that of total charge Q, shared equally).
Net charges: P1: Q/2−2Q/3=−Q/6; P3: Q/2−Q/3=+Q/6. Check: P1+P3=0, as required since they started uncharged and are isolated together.
Insight: the outer faces are not neutral. Ignoring them gives the trap answer −Q/3.
Difficulty: Very hard
Question 3Level 3 · AdvancedSingle correct
The gap of a parallel-plate capacitor (plate area A, separation d, edge effects negligible) is completely filled with a dielectric whose dielectric constant depends only on the distance x from the lower plate as K(x)=2+4x/d for 0≤x≤d. The capacitance of the capacitor is ______. Exactly one option is correct.
Adln34ε0A
Bd4ε0A
Cd3ε0A
Dd2ε0Aln3
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Answer: A
The displacement D=σ is the same across every layer (layers are in series), so E(x)=σ/(ε0K(x)) and
V=∫0dε0(2+4x/d)σdx=ε0σ⋅4dln26=4ε0σdln3.
Hence C=VσA=dln34ε0A≈3.64dε0A.
Check: the value lies between 2 and 6 (times ε0A/d) and below the arithmetic-mean value 4, as it must for layers in series (the inverse of K is averaged, not K).
Trap: 4ε0A/d uses the average of K (correct only for side-by-side regions in parallel).
Difficulty: Very hard
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