An inductor L, capacitor C and two equal resistances R are connected as shown; L = 2R²C. Switch closes at t = 0. At time t₀, the derivatives of branch currents satisfy di₁/dt + di₂/dt = 0. If t₀ = RC ln(k), find k. [See Image.]
Physics · Mixed Chapter
JEE Main Practice

Question

An inductor L, capacitor C and two equal resistances R are connected as shown; L = 2R²C. Switch closes at t = 0. At time t₀, the derivatives of branch currents satisfy di₁/dt + di₂/dt = 0. If t₀ = RC ln(k), find k. [See Image.]

Physics · Mixed Chapter

G1 Physics · Accepted Answer

Step 1 — At switch closure, capacitive and inductive branches obey different first-order laws for their series R elements. Step 2 — Write branch differential equations: L(di₁/dt) + Ri₁ = E and R i₂ + q/C = E, with i₂ = dq/dt, applying initial values i₁(0) = 0 and q(0) = 0. Step 3 — With L = 2R²C, the transient exponents have time constants τ₁ = 2RC and τ₂ = RC. Equate the requested derivatives as indicated. Step 4 — The specified condition occurs at t₀ = RC ln2; k = 2.