Physic Labs

Problem 2

An initially uncharged capacitor C=100 μFC=100\ \mu\mathrm F is connected at t=0t=0 in series with a resistor R=10 kΩR=10\ \mathrm{k\Omega} and an ideal source of emf E=12E=12 V. Let q(t)q(t) be the capacitor charge and i(t)=q˙(t)i(t)=\dot q(t) the current. (a) Find the time constant τ\tau and the long-time charge q∞q_\infty. (b) Write expressions for q(t)q(t) and i(t)i(t). (c) Find when the charge reaches 90%90\% of its final value and the current at that instant.
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τ=RC=(10×103)(100×10−6)=1.00 s,q∞=CE=(100×10−6)(12)=1.20×10−3 C\tau=RC=(10\times10^3)(100\times10^{-6})=1.00\ \mathrm s,\qquad q_\infty=CE=(100\times10^{-6})(12)=1.20\times10^{-3}\ \mathrm C
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The RC equation evolves on the product RCRC, the time constant. After a long time the current vanishes, so the capacitor carries the full emf and its charge is CECE.