Consider a Wheatstone bridge with resistance and capacitance connected as shown. Find the condition on the resistance and the capacitance such that the bridge remains balanced at all times.
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Solution
Suppose that the bridge is balanced ie., VAB=VAD;VBC=VDC Let the current and the charges on the capacitors in the circuit as shown then i1R1=i3R3...(i) q2C2=q4C4...(ii) Consider the charging of the part of the circuit shown alongside. Let q2 be the charge on C2 and i1 be the current in the circuit Then, i1R1+q2C2=ε...(iii);dq2dt+q2R2C2=i1...(iv) Substituting (iv) in (ii) and simplifying dq2dt+q2ReqC2=εR1[1Req=1R1+1R2] ∴q2=εReqC2R1⎡⎢
⎢⎣1−e1ReqC2⎤⎥
⎥⎦=εR2C2R1+R2[1−e−t/ReqC2] Similarly for the other circuit, we have q4=εR4C4R3+R4⎡⎢
⎢⎣1−e1R′eqC2⎤⎥
⎥⎦[1R′eq=1R3+1R4] Now q2C2=q4C4 which leads to R2R1+R2=R4R3+R4;1ReqC2=1R′eqC4orR1R2=R3R4;R1C2=R3C4