A radioactive sample has mean life 20 ms. A capacitor of capacitance 100 μF discharges through resistance R. If the ratio of capacitor charge to radioactive activity remains constant with time, R = 2 × 10ˣ Ω. Find x.
Physics · Mixed Chapter
JEE Mains

Question

A radioactive sample has mean life 20 ms. A capacitor of capacitance 100 μF discharges through resistance R. If the ratio of capacitor charge to radioactive activity remains constant with time, R = 2 × 10ˣ Ω. Find x.

Physics · Mixed Chapter

G1 Physics · Accepted Answer

Step 1 — Activity of a radioactive sample with mean life τ is A(t) = A₀ exp(−t/τ), where τ = 20 ms = 2.0 × 10⁻² s. Step 2 — Charge on the discharging capacitor is q(t) = q₀ exp[−t/(RC)]. Step 3 — For q(t)/A(t) to be constant, the two exponential factors must be identical; therefore RC = τ. Step 4 — Convert C = 100 μF = 100 × 10⁻⁶ F = 10⁻⁴ F. Step 5 — R = τ/C = (2.0 × 10⁻²)/(10⁻⁴) Ω = 2 × 10² Ω. Step 6 — Hence x = 2. Answer: 2.