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`Time Constant of RC CircuitβCurrent in RC CircuitβSource and Reference`

# Time Constant of RC Circuit

Immediately after the RC circuit is connected to battery, `π=0; ππππ=πΌπ` After the capacitor is fully charged, `π=πΆππππ; πΌ=0`

## Current in RC Circuit

The current in RC Circuit is a function of time. Current is defined as Charge βπ passed by per unit time βπ‘ ```πΌ=|π|ππ΄π£βπΌ=βπβπ‘ lim βπ‘β0βπΌ=ππππ‘``` Since current πΌ is the same everywhere in the RC circuit, charge βπ passed by per unit time βπ‘ throughout the circuit piles up at the capacitor πΆ. The voltage ππΆ across the capacitor is ππΆ and voltage ππ across the resistor is πΌπ. Apply voltage loop ```ππππ=ππΆ+ππ=ππΆ+πΌπ=ππΆ+ππππ‘π βππ(π‘)ππ‘=πππππβ1ππΆπ(π‘) ``` Assume π(π‘) has the form π΄πβπ‘/ππΆ+constant, ```π(π‘)=π΄πβπ‘/ππΆ+constantβππ(π‘)ππ‘=β1ππΆπ΄πβπ‘/ππΆ=β1ππΆπ(π‘)βconstant β=1ππΆconstantβ1ππΆπ(π‘)=πππππβ1ππΆπ(π‘) βconstant=πππππΆβπ(π‘)=π΄πβπ‘/ππΆ+πππππΆ ``` Apply boundary conditions, At π‘=0, π=0 `π(π‘β0)=π΄πβ0/ππΆ+πππππΆ=π΄+πππππΆ=0βπ΄=βπππππΆ` Therefore ```π(π‘)=(βπππππΆ)πβπ‘/ππΆ+πππππΆ=πππππΆ1βπβπ‘/ππΆ I(π‘)=ππππ‘=βπππππΆβ1ππΆπβπ‘/ππΆ=ππππππβπ‘/ππΆ ``` ππΆ is the time constant of a RC circuit. When time π‘=ππΆ, the current πΌ drops by a factor of π. `I(π‘)=ππππππβπ‘/ππΆ=ππππππβ1=12.718πππππ=0.37πππππ` By Talor Expansion `πβπ‘/ππΆ=1βπ‘ππΆ+π‘ππΆ212ββ―` When time π‘ and ππΆ are equal, the coefficients of Taylor Expansion equal to 1. In other words, at the time constant ππΆ the current is always reduced to 0.37 of the original current.

## Source and Reference

ID: 200100102 Last Updated: 1/1/2020 Revision: 0

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