rc lab report
Experiment
13
Charging and Discharging Capacitors
1.
Introduction
In this experiment you will measure the rates at which capacitors in series with resistors can be charged and discharged. The time constant RC will be found.
Charging a capacitor.
Consider the series circuit shown in Fig. 1. Let us assume that the capacitor is initially uncharged. When the switch S is open there is of course no current. If the switch is closed at t=0, charges begin to flow and an ammeter will be able to measure a current.
The charges move until the potential across the capacitor plates is equal the potential between the battery's terminals. Then the current ceases and the capacitor is fully charged.
R
R
C
C
Fig. 1. A capacitor in series with a resistor. The left figure represents the circuit before the switch is closed, and the right after the switch is closed at t=0.
The question arises on how does the current in the circuit vary with time while the capacitor is being charged. To answer this, we will apply Kirchhoff's second rule, the loop rule, after the switch is closed q ε − iR −
= 0
C
(1) where q/C is the potential difference between the capacitor plates. We can rearrange this equation as
iR +q/C = ε
(2)
The above equation contains two variables, q and i, which both change as a function of time t. To solve this equation we will substitute for i dq i= dt (3) dq q
R
+ =å
(4)
dt C
This is the differential equation that describes the variation with time of the charge q on the capacitor shown in Fig. 1. This dependence can be found as follows. We will rearrange the equation to have all terms involving q on the left side and those with t on the right side. Then we will integrate both sides dq 1 dt =−
(5)
(q - Cε ) RC q dq
1 t
=−
∫ (q − Cå) RC ∫ dt
0
0 q − Cε t )= − ln (
RC
− Cε
q ( t ) = Cε (1 − e
− t / RC
(6)
(7)
)
(8) where e is the base of the natural logarithm. To find the current
13
Charging and Discharging Capacitors
1.
Introduction
In this experiment you will measure the rates at which capacitors in series with resistors can be charged and discharged. The time constant RC will be found.
Charging a capacitor.
Consider the series circuit shown in Fig. 1. Let us assume that the capacitor is initially uncharged. When the switch S is open there is of course no current. If the switch is closed at t=0, charges begin to flow and an ammeter will be able to measure a current.
The charges move until the potential across the capacitor plates is equal the potential between the battery's terminals. Then the current ceases and the capacitor is fully charged.
R
R
C
C
Fig. 1. A capacitor in series with a resistor. The left figure represents the circuit before the switch is closed, and the right after the switch is closed at t=0.
The question arises on how does the current in the circuit vary with time while the capacitor is being charged. To answer this, we will apply Kirchhoff's second rule, the loop rule, after the switch is closed q ε − iR −
= 0
C
(1) where q/C is the potential difference between the capacitor plates. We can rearrange this equation as
iR +q/C = ε
(2)
The above equation contains two variables, q and i, which both change as a function of time t. To solve this equation we will substitute for i dq i= dt (3) dq q
R
+ =å
(4)
dt C
This is the differential equation that describes the variation with time of the charge q on the capacitor shown in Fig. 1. This dependence can be found as follows. We will rearrange the equation to have all terms involving q on the left side and those with t on the right side. Then we will integrate both sides dq 1 dt =−
(5)
(q - Cε ) RC q dq
1 t
=−
∫ (q − Cå) RC ∫ dt
0
0 q − Cε t )= − ln (
RC
− Cε
q ( t ) = Cε (1 − e
− t / RC
(6)
(7)
)
(8) where e is the base of the natural logarithm. To find the current