6.
Four discharged capacitors C1, C2, C3 and C4 are connected as shown in the figure. A
potential difference is applied between the points X and Y and the system is allowed to
reach steady state. What should the relation between the capacitances of the capacitors be
so that the potential difference between the points x and y is zero?
a. C1/C4 = C2/C3
b. C1/C2 = C3/C4
c. C1 + C4 = C3 + C2
d. 1/C1 + 1/C2 = 1/C3 + 1/C4
Introduction
In many competitive-exam physics questions, a network of four capacitors forms a balanced capacitor bridge where the condition for zero potential difference between two internal nodes must be found. This situation is analogous to the Wheatstone bridge, and understanding the balance condition greatly simplifies solving such capacitor problems.
Circuit explanation and balance condition
The given circuit places C1 and C2 in the upper branch between X and Y, and C4 and C3 in the lower branch between X and Y, with internal nodes x (between C1,C2) and y (between C4,C3). When a steady voltage is applied between X and Y, charges on series capacitors in each branch are equal in magnitude: let the charge on C1 and C2 be q1, and on C4 and C3 be q2.
Potential drop across each capacitor is V=q/C, so the potential at node x relative to X is Vx=q1/C1, while the potential at node y relative to X is Vy=q2/C4. Similarly, measured from Y backward, the potentials of x and y are Vx′=q1/C2 and Vy′=q2/C3. For x and y to be at the same potential (zero potential difference), the potential ratios in the two series branches must match, giving
q1/C1 : q1/C2 = q2/C4 : q2/C3, C1/q1 : C2/q1 = C4/q2 : C3/q2,
which simplifies to
C1/C4 = C2/C3 or C4/C1 = C3/C2.
This is the balance condition of a Wheatstone-like bridge made of capacitors, ensuring no potential difference between the midpoints.
Detailed check of each option
Option (a): C1/C4 = C2/C3
This matches the derived balance condition from equating potentials at x and y in the two series branches.
It is directly analogous to the Wheatstone bridge condition R1/R2 = R3/R4 for zero potential difference between bridge midpoints, with capacitances playing the role of inverse resistances in terms of charge-voltage relations.
Therefore, option (a) is correct.
Option (b): C1/C2 = C3/C4
This option equates the ratio of the two capacitors in the top branch to the ratio of the two capacitors in the bottom branch but in the same left-to-right order.
If this were the correct condition, it would imply C1/C4 = C2/C3 only when also combined with symmetric relationships, but the bridge balance actually requires crossing ratios (left-top with left-bottom vs right-top with right-bottom), not same-branch ratios.
Hence, option (b) does not ensure that potentials at x and y are equal and is incorrect.
Option (c): C1 + C4 = C3 + C2
This condition sums capacitances on opposite sides but ignores how potential divides in series; balance depends on ratios of capacitances, not on their algebraic sums.
Two branches could satisfy this sum condition and still have different voltage divisions across each capacitor, leaving a non-zero potential difference between x and y.
Therefore, option (c) is incorrect.
Option (d): 1/C1 + 1/C2 = 1/C3 + 1/C4
This condition effectively equates total series capacitance of the top branch with that of the bottom branch, since 1/Cseries = 1/Ca + 1/Cb.
Equal series equivalents mean the same overall voltage-to-charge ratio for each branch, but they do not guarantee that the intermediate node potentials (x and y) match; the internal division of voltage still depends on the ratios C1:C2 and C4:C3.
As a result, option (d) is also incorrect.
Key takeaways for exam preparation
A network of four capacitors forming a bridge is balanced (zero potential difference between midpoints) when the product of opposite capacitors is equal, which can be written as C1/C4 = C2/C3 or equivalently C1*C3 = C2*C4.
Recognizing this balanced capacitor bridge condition allows quick elimination of incorrect options that involve simple sums or equal series equivalents instead of correct ratio relationships.
