Charging and discharging a capacitor

| Charging and Discharging a Capacitor |

Figure 1

When a capacitor and a resistor are connected in series to a DC voltage source the charging rate decreases as time passes. The charging current I is given by: đ?‘Ą

đ??ź = đ??źđ?‘œ đ?‘’ −đ?‘…đ??ś

(1)

Where: Io = The current at time t=0 R = The resistance C = The capacitance The voltage V across the capacitor gradually increases according to the following formula: đ?‘Ą

đ?‘‰ = đ?œ€(1 − đ?‘’ −đ?‘…đ??ś )

(2)

Where: É› = The electromotive force (emf) of the source When the charged capacitor is connected to a resistor, it will discharge and the potential difference across its terminals will exponentially decrease:

| Charging and Discharging a Capacitor |

đ?‘Ą

đ?‘‰ = đ?‘‰đ?‘œ đ?‘’ −đ?‘…đ??ś

(3)

Where: V0 = The initial voltage of the capacitor The natural logarithm of Equation (3) is: đ?‘™đ?‘›đ?‘‰ = đ?‘™đ?‘›đ?‘‰đ?‘œ −

And lnV as a function of time is a straight line with slope

1 đ?‘…đ??ś

đ?‘Ą đ?‘…đ??ś

(4)

.

einstein™Tablet+ with MiLAB or Android/iOS Tablet with MiLAB and einstein™LabMate Voltage sensor Box of capacitors Box of resistors 1.5 V battery with battery holder (3) Switch with one pole and two positions (1P2T)

1.

Launch MiLAB (

).

2.

3.

Connect the Voltage sensor to one of the ports on the einstein™Tablet+ or einstein™LabMate.

Assemble the electric circuit as shown in Figures 1 and 2:

Figure 2 4.

5.

6.

Select a capacitor of 1000ď ­F and resistors of 500, 1000 or 2000 ď —.

Attach the red plug of the Voltage sensor to the higher potential terminal of the capacitor, as the Voltage sensor only measures in this polarity.

Make sure that only the Voltage sensor is selected.

| Charging and Discharging a Capacitor |

| Charging and Discharging a Capacitor |

Program the sensors to log data according to the following setup: Voltage Sensor (±25 V)

Voltage (V)

Set As Zero

ON

Rate:

10/sec

Duration:

2 Min

1.

Move the switch to the “b” position (see Figure 2) and wait a few seconds for the capacitor to be completely discharged.

2.

Tap Run ( ) on the main toolbar to begin recording data. A graph displaying the voltage will automatically appear on the screen.

Switch to the “a” position to start charging the capacitor.

When the capacitor is fully charged move the switch to the “b” position to record the discharging process.

Voltage (V)

3.

4.

Figure 3 5.

Tap Save (

) to save data.

For more information on working with graphs see: Working with Graphs in MiLAB. 1.

2.

Zoom in on the discharge part of the graph (see Figure 4).

Use the cursors to select the part of the graph that doesn’t contain zero or near zero voltage values (see Figure 4).

Voltage (V)

| Charging and Discharging a Capacitor |

Figure 4 Tap the Function button (

b.

Tap the Setup button (

). ) next to the Ln function from the Mathematical Functions menu.

c. In the G1 drop down menu select the Voltage data. d. This will give a graph of lnV vs. time (Figure 5).

Next, fit a line to your processed data: a.

Tap the Function button (

).

b. c.

Tap the Linear function from the Curve Fit menu.

The linear fit equation will be displayed below the x-axis.

ln(voltage)

3.

a.

Figure 5 d.

1.

Verify that your results fit the theory (Equation 4) by comparing the slope to its predicted value: 1 |sđ?‘™đ?‘œđ?‘?đ?‘’| = đ?‘…đ??ś

You can repeat the experiment with different resistors and capacitors.

| Charging and Discharging a Capacitor |

2.

You may use a Current sensor simultaneously with a Voltage sensor. You then have many options: a. Plot the derivative of the voltage vs the current:

b.

i.

Tap the Function button (

ii.

Tap the Setup button (

). ) next to the Derivative function from the Mathematical

Functions menu. iii. In the G1 drop down menu select the Voltage data. iv. The potential difference between the terminals of the capacitor is proportional to the charge on them, and since the current is the derivative of the charge it is expected that the voltage derivative will be proportional to the current.

Plot the integral of the current vs the voltage. i.

Tap the Function button (

ii.

Tap the Setup button (

). ) next to the Integral function from the Mathematical

Functions menu. iii. In the G1 drop down menu select the current.