Current Electricity Concept Page - 15

Example
RC circuits at steady state
In the given circuit, the steady state voltage drop across the capacitor C isIn steady state C=0
V=I[r1+r2]
VC=Ir1
         =Vr1r1+r2
Formula
Discharging RC circuit
The differential equation of RC circuit is
dqq=1CRdt
Formula
Charging of an RC circuit
Rdqdt=ϵCqC where ϵ is the emf of the cell
Definition
Time constant in an RC circuit
The time constant of an RC circuit is the time required to charge the capacitor, through the resistor, by 63.2 percent of the difference between the initial value and final value or discharge the capacitor to 36.8 percent.
τ=RC where τ is the time constant and R and C are the values of resistance and capacitance respectively 
Formula
General solution of RC circuit for a capacitor
General solution of RC circuit for a capacitor after time t is given by :
q(t)=q()(q()q(0))etRC
where:
q(t): Charge on the capacitor after time t
q(0): Initial charge on the capacitor
q(): Charge on the capacitor at steady state
RC: Time constant
Note:
The solution remains valid for other parameters of capacitors like voltage, current, etc.
Definition
General solution of RC circuit for a resistor
General solution of RC circuit for a resistor after time t is given by :
i(t)=i()(i()i(0))etRC
where:
i(t): Current in the resistor after time t
i(0): Initial current in the resistor
i(): Current in the resistor at steady state
RC: Time constant
Note:
The solution remains valid for other parameters of resistors like voltage.
Formula
Charge on a capacitor in a charging RC circuit
Charge on a capacitor in a charging circuit is given by the following equation.
q=q0(1etτ)
where q0 is the initial charge of the capacitor and q is the charge at a time t.
Formula
Voltage in a charging/discharging RC circuit
Charging  V(t)=V0(1et/τ)              Vr(t)=V0(et/τ)
 Discharging  Vc(t)=V0(et/τ)           Vr(t)=V0(et/τ)
where τ=RC is the time constant.
Formula
Current in a charging/discharging RC circuit
Charging: i=VRet/τ
Discharging i=V0Ret/τ
where V0 is the initial voltage.

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