Experiment

13

Charging and Discharging Capacitors

1 .

Intro

In this research you will measure the rates when capacitors in series with resistors could be charged and discharged. Enough time constant RC will be located. Charging a capacitor.

Consider the series circuit proven in Fig. 1 . We will assume that the capacitor is initially uncharged. When the change S is open there is of course zero current. In the event the switch can be closed in t=0, fees begin to movement and an ammeter will be able to measure a present. The charges approach until the potential across the capacitor plates is definitely equal the potential between the battery's terminals. Then the current ceases and the capacitor is fully charged.

Ur

R

C

C

Fig. 1 . A capacitor in series with a resistor. The left figure represents the circuit prior to the switch is usually closed, as well as the right after the switch is closed by t=0. The question arises how does the current in the signal vary with time while the capacitor is being charged. To answer this, we is going to apply Kirchhoff's second secret, the loop rule, following the switch is usually closed

queen

Оµ в€’ iR в€’

= 0

C

(1)

where q/C is the potential difference between your capacitor plates. We can piece together this equation as

irgi +q/C = Оµ

(2)

The above equation contains two variables, queen and i, which usually both alter as a function of time big t. To solve this kind of equation all of us will substitute for i

dq

i=

dt

(3)

dq q

3rd there’s r

+ =ГҐ

(4)

dt C

This can be the differential equation that identifies the variance with time of the charge q on the capacitor shown in Fig. 1 ) This dependence can be found as follows. We can rearrange the equation to obtain all conditions involving q on the left side and people with to on the right side. Then simply we will integrate both sides

dq

you

dt

=в€’

(5)

(q - CОµ ) RADIO CONTROLLED

q

dq

1 t

=в€’

в€« (q в€’ CГҐ) REMOTE CONTROL в€« dt

0

0

q в€’ CОµ

to

)= в€’

ln (

RC

в€’ CОµ

queen ( t ) = CОµ (1 в€’ at the

в€’ capital t / RC

(6)

(7)

)

(8)

where electronic is the bottom of the all-natural logarithm. To find the current i(t) we will substitute for q in Eq. 3 solution 8. The derivation of q is

dq

Оµ в€’ t / REMOTE CONTROL

i=

elizabeth

=

dt

R

(9)

where Io = Оµ/R is the primary current in the circuit.

Fig. 2 . Impose vs . period during charging

Fig. a few. Current vs . time during charging

Plots of the fee and the current versus time are shown in Figs. 2 and 3. The charge is usually zero in t=0 and approaches the maximum value of CОµ. The existing has the maximum value of Io=Оµ/R in t=0 and decays tremendously to absolutely no as tв†’ в€ћ. The product RC appears in both equations and has the dimensions of time. The exponents in those

equations must be sizes. RC is named the time frequent of the outlet and is represented by the sign П„. It is the time in that this current inside the circuit offers decreased to 1/e of the initial value. Likewise, in a time П„ the charge improves from absolutely no to C Оµ(1e-1). The across the resistor will change as

в€’ to / RADIO CONTROLLED

в€’ to /П„

Sixth is v R sama dengan iR = Оµ e

= Оµe

(10)

and across the capacitor as

queen

в€’ to / RC

в€’ П„

VC =

) = Оµ (1 в€’ elizabeth t / )

sama dengan Оµ (1 в€’ e

C

(11)

Both capabilities change in time as exponential functions with all the time constant П„=RC. Discharging a capacitor.

Assume that the capacitor in Fig. one particular is totally charged plus the potential across the capacitor can be equal those of the battery. At period t=0 the switch is thrown via a to b in order that the capacitor may discharge through resistor L. Substituting Оµ=0 in Eq. 4 we are able to write the preventing powering equation:

dq

q

R

+

sama dengan 0

dt

C

(12)

The solution in this equation can be

в€’ big t / REMOTE CONTROL

в€’ big t / REMOTE CONTROL

в€’ t/П„

q ( t ) = CОµ e

sama dengan qoe

= qoe

(13)

The current can be acquired by distinguishing Eq. 3

dq

Оµ в€’ to / REMOTE CONTROL

в€’ t/П„

в€’ big t / RC

i(t ) =

electronic

= в€’ Ioe

= в€’ Ioe

= в€’

dt

3rd there’s r

(14)

The minus sign indicates which the direction from the discharge current is in the course opposite for the charging current. Both features, q(t) and i(t), corrosion exponentially while using same period constant П„ = RADIO CONTROLLED.

The potential VR across the resistor is given by

в€’ big t / REMOTE CONTROL

в€’ t /П„

V R (t ) = i (t ) R...