###### Physics

## Measure a Capacitor’s discharge with a Go Direct Voltage Probe

### Brief

A capacitor is a device that stores Charge. This charge is proportional to the potential difference across the capacitor.

V = Q/C

Where C is a proportionality constant known as the *capacitance*. *C* is measured in the unit of the farad, *F*, (1 farad = 1 coulomb/volt)

If a capacitor of capacitance C (in farads), initially charged to a potential V_{0}(volts) is connected across a resistor R (in ohms), a time-dependent current will flow according to Ohm’s law. This situation is shown by the RC (resistor-capacitor) circuit below when the switch is connecting terminals 33 and 34.

As the charge flows, the charge q on the capacitor is depleted, reducing the potential across the capacitor, which in turn reduces the current. This process creates an exponentially decreasing current, modeled by

V(t) = V_{0} e^{-t/RC}

In contrast, when the capacitor is charged, the potential across it approaches the final value exponentially, modeled by

V(t) = V_{0} (1-e^{-t/RC})

The same time constant, RC, describes the rate of charging as well as discharging.

### Objectives

In this experiment, you will

- Measure the potential across a capacitor as a function of time as it discharges and as it charges.
- Measure an experimental time constant of a resistor-capacitor circuit.

### Required items

- Go Direct Voltage probe
- iPad, iPhone, Windows Computer, mac Computer, Chromebook, Android device with the FREE Graphical Analysis 4 Installed
- Electronics circuit kit(such as the Vernier circuit board) that includes different capacitors.

### Circuit Diagram

### Method

Setup the Circuit as shown

On the Go Direct Voltage, press the power button once. The LED will blink red.Launch the Graphical Analysis App. Click Sensor Data Collection. Now select GDX-VOLT from the list (if you have more than one device, select the serial number corresponding to that on the case). The LED on the Voltage probe will now blink Green. Click/Tap* DONE.*

Select ‘Mode’ from the bottom left of the window. Change the sample rate to 100 samples/s and End Collection to 60s (time constants will range between 0.05 and 10 seconds). Click/Tap* DONE. *You are now ready to Collect your first voltage data. Switch the switch so it touching position 32 to charge the capacitor. Press *COLLECT. *After the Capacitor has fully charged switch the switch to position 34. The Capacitor will now discharge. The data will automatically stop collecting after 60 seconds.

### Analysis

On completion the graph will auto-scale. Click on the windowed icon, top right and select ‘Table’. Click on the ellipsis in the Potential column header and choose ‘Add Calculated Column’. In the ‘Expression’ box choose ‘A ln(X), leaving A as 1.

Switch back to 1 graph, using the window icon, top right. Click in the x-axis label and select both Potential V and Natural Log V. For the time after discharge began, the Voltage should be given by:

V(t) = V_{0} e^{-t/RC}

So the natural log form gives us:

*ln(V(t)) = – (1/RC)t + ln(V**0)*

If the section of the line corresponding to the discharge period is straight, then we can conclude that the graph is exponential. To determine the gradient, which is equal to -1/*RC*, click and drag over the straight section of the ln (*V*) graph. Click on the ‘graph tools’ icon, bottom left, and choose ‘view statistics’. Record the values Δ*Y* and Δ*X* and hence calculate the gradient and the value *RC*.

Repeat the experiment for different values of *R* and *C*, recording the calculated time constant for each pair

### Extension

Measure the resistance of each resistor with a multimeter.

For each RC pair, calculate the capacitance using the measured values of RC and R.

*C = T/R*

Compare the measured value of capacitance to that stated on the capacitor and compare to the claimed tolerance of the component.

### Example Data

Here is some example Data