Practical Electricity and Measurement for Electrodynamics

EDL01: Practical Electricity and Measurement for Electrodynamics

Laboratory purpose. Before using instruments to investigate electric fields, magnetic fields, induction, and electromagnetic waves, we must understand how a measurement changes the circuit being measured. A voltmeter draws current, an ammeter introduces a small voltage drop, a signal generator has an output impedance, and an oscilloscope probe can add capacitance. These effects are small in some experiments and decisive in others.

This entry develops a reusable measurement method through a sequence of low-voltage experiments. It is an experimental companion to the EDL00 roadmap and the PhysicsLibrary Electromagnetism series. It is not a substitute for circuit theory or for the field formulation of electrodynamics.

At the end of the article, the reader should be able to measure a resistor’s current-voltage relation, determine whether an instrument is loading a circuit, infer a source’s voltage and resistance, observe an RC transient, and explain why the oscilloscope ground clip must be connected deliberately.

1 Laboratory questions and learning objectives

The central question is:

(1)

We will learn to distinguish the unloaded prediction, the loaded prediction, and the observed quantity. The repeated experimental cycle is

(2)

Specific objectives are to

• distinguish current through a branch from the potential difference across it;
• verify I = V∕R using measured, rather than nominal, resistor values;
• derive and observe the effect of finite voltmeter input resistance;
• infer the input resistance of a digital multimeter (DMM);
• infer the open-circuit voltage and output resistance of a signal source;
• observe finite probe capacitance and compare 1× with 10× probing;
• measure an RC time constant and compare it with RC;
• identify common-ground connections and avoid accidental short circuits;
• report an uncertainty budget, rather than treating display digits as exact.

2 Equipment options

The entry is divided into a basic DMM track and an optional oscilloscope track. The instrument models below are examples, not required brands.

2.1 Minimum track: low-voltage DC

• An adjustable current-limited DC bench supply, used at 0 to 5 V; or a low-voltage battery pack with a series protection resistor.
• One DMM with voltage and resistance functions; two DMMs are convenient for simultaneous voltage and current measurements.
• Breadboard, insulated leads, and resistors near 470 Ω, 10 kΩ, 100 kΩ, 1 MΩ, and 2.2 MΩ.
• A capacitor near 47 μF for a slow, DMM-visible transient.
• A notebook or the accompanying CSV data sheets.

2.2 Extended track: time-dependent measurements

• A function generator with low-frequency sine and square-wave outputs.
• A two-channel oscilloscope and compatible probes, preferably with selectable 1× and 10×operation.
• Additional 100 Ω and 1 kΩ resistors, and a capacitor near 100 nF.
• BNC breakout leads or an approved BNC-to-breadboard adapter.

Measure every resistor with Power disconnected before using it. Resistor color bands and nominal capacitance labels do not establish the true values. The manufacturer’s specifications for a particular DMM, scope, and generator should be kept with the lab record.

3 Safety and setup

All circuit experiments in this entry are limited to an isolated, low-voltage bench source. Never connect the circuits to a wall outlet, a mains-derived unisolated circuit, a high-voltage supply, or a high-energy battery pack. Do not use an oscilloscope to probe mains wiring.

1. Disable the power output before changing wires or meter functions.
2. Start with a 0 V setting and a conservative supply current limit. For a 470 Ω load at 5 V, the expected current is about 10.6 mA; a current limit near 20 to 30 mA is sufficient for that particular experiment.
3. Check the power dissipated in a resistor using P = V ²∕R and its manufacturer’s rating.
4. Place an ammeter in series, never directly across a powered source.
5. Place a voltmeter in parallel with the two nodes of interest.
6. Verify scope-probe ground connections before enabling the source. For a conventional earth-grounded bench oscilloscope, channel ground clips are connected to one another and to protective earth.
7. Never defeat the protective-earth connection to make a measurement.
8. Discharge capacitors through a suitable resistor before rewiring. Observe electrolytic capacitor polarity and voltage rating.

A meter’s current input may be fused and may have a different terminal from its voltage input. After measuring current, move the lead back to the voltage terminal before measuring a potential difference.

4 Electrical quantities and the measuring circuit

A node is a collection of points connected by Conductors that are modeled as having negligible voltage drop. Node voltage V _A is a potential measured relative to a chosen reference node. The voltage across an element between nodes A and B is

(3)

A branch current is the rate at which charge passes a selected cross-section:

(4)

The sign of I depends on the reference direction. The passive sign convention takes positive current as entering an element’s positive-voltage terminal. For an ohmic resistor,

(5)

Voltage is a difference between nodes. Current is measured through a branch. Consequently, the two measurements require different instrument connections.

A real voltmeter has a finite input resistance R_M. It draws approximately

(6)

for DC operation. A real ammeter has nonzero internal resistance and introduces an ammeter burden voltage, often written

(7)

where R_A is the effective current-path resistance. DMM specifications may instead give a maximum burden per measurement range. In high-resistance or low-voltage circuits, both loading mechanisms may matter.

5 Experiment 1: Ohm’s law from independently measured data

5.1 Prediction

Connect a resistor R to an adjustable DC source. If it is approximately ohmic over the tested range,

(8)

Thus, a plot of measured current I against measured resistor voltage V _R should be linear, with slope

(9)

The slope-inferred resistance is

(10)

This is better than comparing current with the bench supply’s displayed voltage, since an ammeter burden or supply-lead drop may make that displayed voltage different from V _R.

5.2 Construction and measurement

1. With power off, measure the actual value of a nominal 470 Ω resistor and record the DMM range.
2. Wire the resistor in series with the DMM current input. Connect a second DMM across the resistor for voltage. With one DMM, disable the source before reconnecting the meter, then repeat the setting to obtain voltage. Since reconnecting changes the current-path burden, a second meter or a separately measured sense resistor gives a more consistent two-quantity measurement.
3. Set the current limit conservatively and raise the supply through approximately 1, 2, 3, 4, and 5 V.
4. Record actual V _R and I at each setting, not only the source display.
5. Plot I against V _R. Fit a straight line and compare 1∕S with the independently measured resistor value.
6. Reverse the resistor connections. Repeat one point. For a simple resistor, the measured positive V∕I ratio should not depend on orientation.

A resistor measured at 470 Ω and carrying 10.0 mA should have a voltage of

(11)

This number is an illustrative prediction, not a supplied experimental result.

5.3 Inference and perturbation

Check whether the fitted intercept is consistent with zero within experimental resolution. A significant intercept may indicate offset, an inappropriate measurement range, or recording error. Repeat with a different resistance. An Ohm’s-law experiment is complete only after the changed component has produced the expected change in slope.

6 Experiment 2: an unloaded and loaded voltage divider

6.1 Prediction without a meter

Two resistors R₁ and R₂ connected in series across a DC source V _S form a voltage divider. Taking the bottom node as reference, the output without a measurement load is

(12)

For R₁ = R₂, the output is V _S∕2. But an attached voltmeter provides another path from the output node to the reference node.

6.2 Loaded prediction

With input resistance R_M, the lower leg becomes

(13)

Therefore,

(14)

The divider can also be described by its Thevenin parameters:

V th	= V S,	(15)
Rth	= .	(16)

The voltage observed with the meter is

(17)

We can solve this expression for the unknown meter input resistance:

(18)

This inference becomes sensitive to measurement errors if the loading is too small. Choose a divider whose R_th is a substantial fraction of R_M while keeping the DMM input within its specified measurement range.

6.3 Procedure

1. Assemble a divider with R₁ = R₂ = 10 kΩ and V _S ≃ 5 V. Record the actual values and measured V _S.
2. Measure the output and compare with the unloaded prediction.
3. Replace both resistors with values near 2.2 MΩ. Keep the supply voltage unchanged and repeat.
4. Calculate R_M using measured resistances and voltages.
5. Change the DMM range, if possible, and repeat. Do not assume that the input resistance is constant on every range or in a high-impedance mode.

6.4 Worked illustrative estimate

Let V _S = 5.00 V, R₁ = R₂ = 2.20 MΩ, and R_M = 10.0 MΩ. Then

(19)

The measured-node prediction is

(20)

The meter depresses the output by approximately 0.248 V. This is nearly 10% of the unloaded value, even when the DMM behaves perfectly according to its specification.

7 Thevenin model: an experimentally inferred source

A Thevenin equivalent is a two-terminal model with an ideal open-circuit voltage V _th in series with resistance R_th. For a resistive load R_L, the terminal voltage is

(21)

The model is an experimentally useful two-terminal description for a linear network at DC. For an AC generator, R_th is a practical approximation to the output impedance over a specified frequency range. Do not assume the impedance stays purely resistive at every frequency.

With two load values (R_L1,R_L2) and observed terminal voltages (V ₁,V ₂), we have

V 1	= V th,	(22)
V 2	= V th.	(23)

Equating the resulting expressions for V _th gives

(24)

The source voltage can then be recovered from either load:

(25)

The two-load formula is especially helpful when a true open-circuit measurement would be distorted by the measuring instrument.

8 Experiment 3: measure a function generator’s source resistance

8.1 Prediction

A common laboratory generator has a nominal 50 Ω output impedance. This is a typical instrument design, not a universal specification. The voltage shown on its display may be calibrated for a specified load (usually 50 Ω or high impedance). Accordingly, the displayed amplitude is not necessarily the open-circuit terminal amplitude.

Let the actual open-circuit sine-wave amplitude be A_oc. For a resistive load,

(26)

Measure the same amplitude convention, preferably peak-to-peak voltage V _pp, in all runs. Do not mix peak, RMS, and peak-to-peak amplitudes. For a zero-offset sine wave,

(27)

8.2 Procedure: optional oscilloscope track

1. Choose a sine wave near 1 kHz. Start with a small amplitude, such as about 1 V_pp when the display is configured for high-impedance loading. Keep output offset at zero.
2. Inspect the generator’s amplitude-load setting and document it. Connect the scope ground to the generator’s signal return only.
3. Measure the output amplitude with a high-impedance 10× probe and no added load. This is an approximation to the open-circuit amplitude.
4. Attach a measured 1 kΩ resistor from signal to return and record its terminal V _pp.
5. Replace the load with a measured 100 Ω resistor and record V _pp again. Verify that the generator is rated to drive it.
6. Infer R_S using both the open-circuit approximation and the two-load formula. Compare the results.
7. Repeat at a second, moderate frequency to test whether the inferred resistance is approximately stable.

If R_S = 50 Ω and the open-circuit amplitude is 1.00 V_pp, then the predicted amplitudes are

A1k	= 1.00 ≃ 0.952 Vpp,	(28)
A100	= 1.00 ≃ 0.667 Vpp.	(29)

These are examples, not substitute data. The measured resistor values and scope readouts belong in the lab record.

9 Experiment 4: oscilloscope probe loading

A probe does not merely transport a voltage to the oscilloscope. Its input is reasonably modeled, over a restricted frequency range, by an effective resistance R_P in parallel with an effective capacitance C_P :

(30)

The source resistance R_S and probe capacitance create a low-pass effect. If resistive input loading is negligible,

(31)

Its approximate corner frequency is

(32)

A common bench scope input is about 1 MΩ with appreciable parasitic capacitance; a suitable 10× passive probe may present about 10 MΩ and a much smaller effective capacitance. The actual values depend on the scope, probe, and attenuation switch. An uncompensated passive probe can distort square-wave edges even when its DC attenuation is correct.

9.1 Procedure

1. Connect the generator through a measured 100 kΩ series resistor to a scope input. Use a small sine wave.
2. Begin at about 1 kHz. Compare source-side and measurement-node amplitudes when a second scope channel is available.
3. Repeat around 10, 20, and 50 kHz, keeping the source amplitude fixed.
4. Compare the 1× connection with a correctly configured 10× probe. Match the probe attenuation setting in the scope.
5. Compare measurements with the model using the manufacturer’s approximate R_P and C_P . Note that cable capacitance and compensating networks can modify the simple model.

Do not use a long, unmatched loose wire as an assumed equivalent to a calibrated scope probe. A 10× probe must be compensated according to the scope manufacturer’s instructions before interpreting edge shape.

10 Experiment 5: RC charging and discharging

10.1 Prediction from charge continuity

For a resistor R in series with a capacitor C, charge continuity and Ohm’s law give

(33)

For a step from zero to constant V ₀ with V _C(0) = 0,

(34)

For discharge through R from initial voltage V ₀,

(35)

The RC time constant is

(36)

At t = τ, the ideal charging response reaches 1 − e⁻¹ ≃ 0.632 of its final voltage, while the discharging response falls to e⁻¹ ≃ 0.368 of its initial value.

10.2 Track A: slow response with a DMM

1. Measure a 100 kΩ resistor and a nominal 47 μF capacitor. Check polarity for an electrolytic type.
2. Connect R and C as a low-voltage series charging circuit. Connect the DMM across the capacitor.
3. Predict τ ≃ 4.7 s using nominal values.
4. Apply about 5 V and record V _C at t = 0, 2, 5, 10, 15, and 20 s or at comparable intervals.
5. Disable and disconnect the source, discharge through the resistor, and record the voltage-decay curve. Do not directly short the capacitor.
6. Estimate τ from the 63.2% or 36.8% time, noting the DMM update rate and manual timing uncertainty.

For this arrangement, the DMM’s finite input resistance is in parallel with the capacitor. It can shift the steady-state voltage and the inferred transient time constant. Check whether R_M is sufficiently large compared with R or incorporate it into the circuit model.

10.3 Track B: function generator and oscilloscope

1. Use a resistor near 10 kΩ and a capacitor near 100 nF, with predicted τ ≃ 1.0 ms.
2. Apply a low-frequency square wave, about 20 to 25 Hz, with a safe amplitude and offset that keep the capacitor within its ratings.
3. Probe the capacitor using 10× mode. Record the voltage just after the edge and several points through the transition.
4. Determine τ from the 63.2% rise, the 36.8% decay, or from t₉₀ − t₁₀ ≃ 2.197τ for an ideal rising exponential.
5. Replace R with approximately twice its original value. Check whether the observed time constant approximately doubles.

A function generator contributes additional source resistance. If its output resistance is relevant to the experiment, replace R in the simple model by R + R_S, after verifying the topology and loading.

11 Experiment 6: understand oscilloscope ground

On a typical earth-grounded two-channel bench oscilloscope, the ground clips on both channels are electrically common. They are also normally connected to the protective-earth conductor through the instrument chassis. The probe tip measures its voltage relative to this ground.

For an ordinary low-voltage series RC circuit, connect both scope ground clips to the circuit return. Connect channel tips to the source and capacitor nodes. The difference between channels may be computed using a verified scope math function, such as V _R = V _CH1 −V _CH2, provided both channels share the same reference and their settings match. This does not require connecting a ground clip to the resistor’s upper node.

11.1 Safe demonstration

1. With all outputs disabled, use a continuity function to check whether the scope channel ground shells are common. Verify behavior with the instrument manual.
2. Build a divider or RC circuit using only the approved low-voltage source and a single designated return node.
3. Attach both probe grounds to that return node.
4. Observe the separate node voltages and, if available, their difference using the scope math channel.
5. Sketch a proposed but unconnected ground lead attached to an intermediate node. Identify which component would be bypassed. Do not make the short to demonstrate it physically.

Battery-operated isolated scopes and differential probes can have different reference behavior. Isolation must be verified from their manuals; it must not be assumed from appearance or portability.

12 Quantifying the uncertainty

An experimental value should be reported with at least an estimated measurement uncertainty and a description of significant systematic effects. For a derived resistance R = V∕I, independent small uncertainties give

(37)

If the same instrument introduces correlated errors, this expression is only an approximation. Manufacturer accuracy statements can include a percent of reading plus a number of counts. Instrument resolution is not identical to accuracy.

For a capacitance inferred from τ = RC,

(38)

so, for independent uncertainties,

(39)

For the inferred DMM input resistance,

(40)

errors become magnified when V _M is nearly equal to V _th. A divider selected to make loading visible therefore produces a more stable inference than one with negligible observed loading.

Each experiment should report

• the measured component values and instrument model or range;
• the theory prediction using those values;
• observed voltages, currents, or transient times;
• an instrument-loading estimate;
• at least one deliberate change to the apparatus;
• and an explanation of the dominant uncertainty.

13 Data processing and lab deliverables

The accompanying package includes blank CSV worksheets for experiments 1 through 5, an example prediction worksheet, and a small Python analysis script. The example values in the prediction worksheet are synthetic illustrations, not measured observations. Keep your raw data separate from derived results.

For each experiment, include a circuit sketch, a table of raw measurements, one graph with axes and units, at least one fitted or inferred parameter, and a concise physical interpretation. Include an explanation when the measured result is inconsistent with the ideal model.

14 Concept questions and experimental challenges

1. A voltmeter reads 2.25 V across a nominally equal high-resistance divider connected to a 5.00 V source. Why is that not necessarily evidence that either divider resistor is defective?
2. Why does measuring the voltage at the DMM terminals matter in an Ohm’s-law experiment using a series ammeter?
3. A generator gives about twice the peak-to-peak voltage into a high-impedance input as into a 50 Ω termination. What source model does this suggest, and what limitations remain?
4. A passive scope probe shows a rounded square-wave edge. Give two possible physical causes unrelated to the device under test.
5. Why can connecting channel 1 ground to one RC node and channel 2 ground to a different RC node alter the circuit even before measuring anything?
6. An RC transient is measured with a slow DMM and then with a scope. What sources of uncertainty dominate in the two cases?
7. In the divider experiment, propose two changes that would reduce loading, and explain the tradeoff between power consumption and measurement accuracy.

15 Answers and physical interpretation

1. The DMM has a finite input resistance in parallel with the divider’s lower resistor; this changes its output. Use a loaded-divider calculation.
2. The ammeter has finite burden and the source leads have nonzero resistance; the resistor voltage is the quantity that belongs in V _R = IR.
3. A source near 50 Ω in series with an ideal voltage may explain the half-amplitude observation. Display loading settings and frequency-dependent impedance still need checking.
4. Probe capacitance and incorrect passive-probe compensation can round or distort edges, in addition to the circuit’s genuine bandwidth limitation.
5. On a common-ground scope, the ground clips electrically join the nodes and may bypass a component or short the source.
6. Manual timing and display update speed often dominate the slow experiment. Probe loading, trigger placement, bandwidth, and sampling can matter in the scope experiment.
7. Reduce divider resistance or use a higher-impedance measurement system. Lower divider resistance reduces fractional meter loading but increases supply current and resistor dissipation.

16 Connection to the rest of electrodynamics

The instrument is part of the electromagnetic system, not an invisible observer. This practical fact links the circuit experiments to the later field description. Current is transported charge, terminal voltage is a potential difference, a capacitor stores electric-field energy, and source impedance controls energy transfer. As measurement frequency rises, capacitance, inductance, spatial distribution, and propagation increasingly matter. The simple lumped-element models used here will be tested and refined in later EDL entries.

The next laboratory, EDL02, begins electrostatics and charge-induction experiments. EDL01 provides the measurement discipline needed to distinguish physical electric-field effects from detector and instrument artifacts.

17 References

1. PhysicsLibrary, EDL00, Home Electrodynamics Laboratory: Roadmap, Bench Setup, Measurement Practice, and Griffiths Companion.
2. D. J. Griffiths, Introduction to Electrodynamics, 5th ed., Cambridge University Press. Unofficial experimental companion.
3. Relevant operating manuals and input/output specifications for the DMM, oscilloscope, probes, DC supply, and function generator actually used.