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Time Constant of RC Circuit
 Current in RC Circuit
 Source and Reference

Time Constant of RC Circuit

image Immediately after the RC circuit is connected to battery, 𝑄=0; π‘‰π‘’π‘šπ‘“=𝐼𝑅 After the capacitor is fully charged, 𝑄=πΆπ‘‰π‘’π‘šπ‘“; 𝐼=0

Current in RC Circuit

The current in RC Circuit is a function of time. Current is defined as Charge βˆ†π‘„ passed by per unit time βˆ†π‘‘ 𝐼=|π‘ž|𝑛𝐴𝑣⇒𝐼=βˆ†π‘„βˆ†π‘‘
lim βˆ†π‘‘β†’0⇒𝐼=𝑑𝑄𝑑𝑑
Since current 𝐼 is the same everywhere in the RC circuit, charge βˆ†π‘„ passed by per unit time βˆ†π‘‘ throughout the circuit piles up at the capacitor 𝐢. The voltage 𝑉𝐢 across the capacitor is 𝑄𝐢 and voltage 𝑉𝑅 across the resistor is 𝐼𝑅. Apply voltage loop π‘‰π‘’π‘šπ‘“=𝑉𝐢+𝑉𝑅=𝑄𝐢+𝐼𝑅=𝑄𝐢+𝑑𝑄𝑑𝑑𝑅
⇒𝑑𝑄(𝑑)𝑑𝑑=π‘‰π‘’π‘šπ‘“π‘…βˆ’1𝑅𝐢𝑄(𝑑)
Assume 𝑄(𝑑) has the form π΄π‘’βˆ’π‘‘/𝑅𝐢+constant, 𝑄(𝑑)=π΄π‘’βˆ’π‘‘/𝑅𝐢+constant⇒𝑑𝑄(𝑑)𝑑𝑑=βˆ’1π‘…πΆπ΄π‘’βˆ’π‘‘/𝑅𝐢=βˆ’1𝑅𝐢𝑄(𝑑)βˆ’constant  =1𝑅𝐢constantβˆ’1𝑅𝐢𝑄(𝑑)=π‘‰π‘’π‘šπ‘“π‘…βˆ’1𝑅𝐢𝑄(𝑑) β‡’constant=π‘‰π‘’π‘šπ‘“πΆβ‡’π‘„(𝑑)=π΄π‘’βˆ’π‘‘/𝑅𝐢+π‘‰π‘’π‘šπ‘“πΆ Apply boundary conditions, At 𝑑=0, 𝑄=0 𝑄(𝑑→0)=π΄π‘’βˆ’0/𝑅𝐢+π‘‰π‘’π‘šπ‘“πΆ=𝐴+π‘‰π‘’π‘šπ‘“πΆ=0⇒𝐴=βˆ’π‘‰π‘’π‘šπ‘“πΆ Therefore 𝑄(𝑑)=(βˆ’π‘‰π‘’π‘šπ‘“πΆ)π‘’βˆ’π‘‘/𝑅𝐢+π‘‰π‘’π‘šπ‘“πΆ=π‘‰π‘’π‘šπ‘“πΆ1βˆ’π‘’βˆ’π‘‘/𝑅𝐢
I(𝑑)=𝑑𝑄𝑑𝑑=βˆ’π‘‰π‘’π‘šπ‘“πΆβˆ’1π‘…πΆπ‘’βˆ’π‘‘/𝑅𝐢=π‘‰π‘’π‘šπ‘“π‘…π‘’βˆ’π‘‘/𝑅𝐢
𝑅𝐢 is the time constant of a RC circuit. When time 𝑑=𝑅𝐢, the current 𝐼 drops by a factor of 𝑒. I(𝑑)=π‘‰π‘’π‘šπ‘“π‘…π‘’βˆ’π‘‘/𝑅𝐢=π‘‰π‘’π‘šπ‘“π‘…π‘’βˆ’1=12.718π‘‰π‘’π‘šπ‘“π‘…=0.37π‘‰π‘’π‘šπ‘“π‘… By Talor Expansion π‘’βˆ’π‘‘/𝑅𝐢=1βˆ’π‘‘π‘…πΆ+𝑑𝑅𝐢212βˆ’β‹― When time 𝑑 and 𝑅𝐢 are equal, the coefficients of Taylor Expansion equal to 1. In other words, at the time constant 𝑅𝐢 the current is always reduced to 0.37 of the original current.

Source and Reference

https://www.youtube.com/watch?v=dPGTENyJO9k&list=PLZ6kagz8q0bvxaUKCe2RRvU_h7wtNNxxi&index=17

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

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