Question

An uncharged capacitor and a resistor are connected in series to a source of emf. If e m f = 9.00 V, C = 25.0 µF, and R = 100 Ω, find the following:

(a) the time constant of the circuit (s)

(b) the maximum charge on the capacitor (µC)

(c) the charge on the capacitor after one time constant (µC)

Answer 1

see the digram :

LP^

given : V = 9V

C = 25 x 10-6 F

R = 100 ohm.

at time t = 0, charge on capacitor =q = 0.

at any time t , charge on capacitor = q, and current in the circuit is i.

also, i = dq/dt

So, at any time instant t,

applying kirchoff's law:

V=2+R

$\Rightarrow \int_{o}^{q}\frac{dq}{CV-q} = \int_{o}^{t}\frac{dt}{RC}$

solving this equation, we get charge q on capacitor as a function of time :

----- (i)

the term RC is called time constant.

a) time constant = RC = 100 x 25 x 10-6 = 25 x 10-4 s.

b) the charge on capacitor is maximum when $e^{-\frac{t}{RC}}$ is minimum.

so, if we put t = infinity, then $e^{-\frac{t}{RC}}$ = 0.

Hence, maximum charge on capacitor = CV = 25 x 10-6 x 9 = 225 x -6 C = 225 micro coulomb.

c) after t = 1 time constant, i.e; t = RC,

charge on capacitor :

q=CV (1 - e RC

$\Rightarrow q = CV(1-e^{-1})$

=> q = 142.22 micro coulomb.