PHYS112 : Labs

9 Inductance

In addition to reading this assignment, read Appendix C: Oscilloscope Guide. You may also need to refer to Appendix A on uncertainties.

Introduction

An inductor is a coil of wire which stores energy in a magnetic field when it carries a current. The inductance of an inductor is defined as the magnetic flux through the inductor per unit current

$\begin{displaymath} L = \frac{\Phi^\mathrm{mag}}{i} \end{displaymath}$

(25)

Any change in the current through an inductor leads to a change in the magnetic field it produces. This in turn leads to a change in the magnetic flux through the inductor which, by Faraday's law, produces an induced emf in the inductor.

**Figure 20:** (a) A series LR circuit. (b) The potential difference across the inductor after the switch is closed.
$\includegraphics{LR-voltage.eps}$

The moment the switch in the circuit shown in Figure 20 is closed, there is no current flowing, and the voltage across the inductor is the same as the emf $\mathcal{E}$ of the voltage source. The current immediately starts to increase from zero, changing rapidly at first, and leveling off as it reaches its steady-state value of $\mathcal{E}/R$ . The potential difference across the inductor decays exponentially with the rate of change of the current as

$\begin{displaymath} V_L = \mathcal{E} e^{-t/\tau_L} \end{displaymath}$

(26)

where $\tau_L$ is the time constant of the LR circuit given by

$\begin{displaymath} \tau_L = \frac{L}{R} \end{displaymath}$

(27)

This is the time it takes to fall to a fraction of its initial charge. As you discovered in measuring capacitance in Lab 7, it is easier in practice to measure the half-time $t_{1/2}$ of an exponential decay than its time constant. The time constant $\tau_L$ is related to the half time $t_{1/2}$ via

$\begin{displaymath} \tau_L=\frac{t_\frac{1}{2}}{ln(2)} \end{displaymath}$

(28)

Just as the capacitance of a capacitor depends on the geometry of its conductors, the inductance of an inductor depends on the geometry of its wire loops. The inductance of a solenoid of cross sectional area , length $\ell$ , and having turns is given by

$\begin{displaymath} L = \frac{\mu_o N^2 A}{\ell} \end{displaymath}$

(29)

You will use an oscilloscope to measure the inductance of the solenoid you worked with in Lab 8. You will also investigate the behavior of an LR filter. First, you will have a chance to familiarize yourself with many of the controls and functions of an oscilloscope (see Appendix C).

Experiments and Analysis

Introduction to the Oscilloscope

Connect the alligator leads from channel 1 of the oscilloscope across the output of the function generator.
Set the function generator to produce a sinusoidal wave form.
Set the triggering source to CH1.
Trigger on a positive slope.
Adjust the horizontal and vertical scale and position knobs and triggering level knob so that the signal is stable, nice and big, and centered in the display. If you run into trouble, ask for help.
Experiment with the triggering level and slope controls until you understand what they do. Under what circumstances does triggering fail, giving a display like Figure 32 of Appendix C?
Disconnect the alligator leads from the function generator and have someone hold one lead in each hand. Measure the frequency of the signal. What could this be?

Measuring Inductance

**Figure 21:** (a) A circuit in which a function generator with internal resistance and a square wave form drives current back and forth through an inductor with internal resistance in series with a resistor . (b) A qualitative plot of the potential difference across the inductor vs. time.
$\includegraphics{LR-circuitb.eps}$

Measure and record the resistances of the solenoid and of the resistor with the DMM.
You should find the internal resistance of the function generator on a label next to its output terminals. Record them. (This can't be measured directly with the DMM.)
Construct the circuit shown in Figure 21. It is essential to connect the black leads from the function generator and oscilloscope to the opposite end of the solenoid from the resistor , because they are grounded.
Connect the oscilloscope in parallel with the solenoid.
Set the function generator to produce a square wave form.
Adjust the vertical and horizontal scales and positions of the oscilloscope to display the voltage across the inductor.
Adjust the frequency of the function generator to obtain a signal like that shown in Figure 21.
Note that the potential difference across the inductor decays to a small constant value instead of zero. This is the voltage drop across the internal resistance of the solenoid.
Measure the half-time of the decay of the induced emf in the solenoid. To get the best precision possible, adjust the horizontal and vertical scales so that a single decay fills the oscilloscope display.
One strategy:
- Use the vertical position knob to place the baseline of the decay (the lower dashed line in Figure 21) at the bottom of the display.
- Use a combination of the outer VOLTS/DIV knob and inner variable scale knob to stretch the signal vertically so that the maximum voltage (the upper dashed line in Figure 21) is at the top of the display so that the total drop in potential $\Delta V$ across the inductor spans the vertical scale of the display.
- The half-time of the decay is the time interval between the peak and the point at which the signal crosses the center of the display.
- For best precision, use the TIME/DIV knob to stretch the signal as much as you can before making the half-time measurement.
Determine the inductance of the coil and its uncertainty from your half-time measurement. Remember to use the sum of the resistances of the resistor and the internal resistances of the function generator and solenoid in your calculations!
Use Eq. 29 and your measured inductance to determine the number of turns in the solenoid and the associated uncertainty.

Before You Leave Lab

Discuss with your instructor preliminary answers to the questions below.

Hand In ...

... answers to the following.

Under what circumstances could you get the display to look like the mess in Figure 32 of Appendix C? Explain why triggering fails.
What do you think is the source of the signal you measured across a person's hands? Explain.
Give your results for the inductance and the number of turns of the solenoid with uncertainties. Show the calculations you did to get them.