We know that voltage across a capacitor is given by
But there is this other formula
How this second formula was deducted mathematically from the first one? or from another one?
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\$\begingroup\$ in the second formula, what is R, \$V_s\$ and \$V_c\$? You need to explain it to us \$\endgroup\$ – FrancoVS Jun 26 '18 at 11:18
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\$\begingroup\$ @FrancoVS - Vc should be voltage on capacitor. Vs will be the source voltage. \$\endgroup\$ – Whiskeyjack Jun 26 '18 at 11:29
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\$\begingroup\$ Do you know how to solve an ODE? \$\endgroup\$ – Spehro Pefhany Jun 26 '18 at 11:31
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Let I_r be current in the resistor = Vc/R at any time.
Vc = Q/C from the basic definition of capacitance.
dQ/dt = -I_r = -Vc/R From the basic definition of current.
dQ/dt = -Q/CR
Apply initial conditions, Q at time 0 = Vs * C and do trivial calculus, then substitution to get Vc in place of Q.
