FRQ 3 – Experimental Design

Overview

• Question 3 of 4 in Section II
• Worth 10 points (12.5% of your total exam score)
• Suggested time: 25-30 minutes
• Calculator essential for data analysis
• Tests skills: Designing E&M experiments (1.B, 3.A), Analyzing electrical data with calculus (2.B, 2.D), Circuit analysis and linearization (2.C)

The Experimental Design and Analysis question in E&M assesses your ability to create sophisticated procedures for measuring electromagnetic quantities and analyze data using advanced techniques. You'll design experiments to determine properties like resistivity, capacitance, inductance, or magnetic field strength, often requiring careful circuit design or field mapping procedures. The analysis frequently involves exponential fits, phase relationships, or extracting parameters from non-linear behaviors. This question bridges theoretical E&M with practical measurement challenges.

Strategy Deep Dive

E&M experimental design requires understanding both theoretical relationships and practical measurement limitations.

Designing Experiments with Electrical Measurements

E&M experiments often involve time-varying quantities requiring sophisticated approaches:

• Direct measurements: Multimeters for voltage/current/resistance
• Oscilloscope measurements: Time-varying signals, phase relationships
• Field mapping: Hall probes, compass arrays, test charges
• Indirect determination: Measuring accessible quantities to infer inaccessible ones

Consider what's actually measurable in a lab. You can't directly measure electric field strength, but you can measure force on a test charge. You can't see magnetic field lines, but you can map compass deflections.

Circuit Design for Parameter Extraction

Many E&M experiments involve clever circuit design:

Measuring Capacitance:

• RC circuit with known R
• Measure voltage decay: V(t) = V₀e^(-t/RC)
• Plot ln(V) vs. t to get slope = -1/RC
• Extract C from slope

Measuring Inductance:

• RL circuit with square wave input
• Measure current rise time to 63.2% (one time constant)
• τ = L/R gives inductance
• Or use LC resonance: f = 1/(2π√LC)

Internal Resistance:

• Measure terminal voltage vs. load current
• Plot V vs. I gives slope = -r (internal resistance)
• Y-intercept gives EMF
• Need variable load resistance

Field Mapping Strategies

Electromagnetic fields require spatial measurement strategies:

Electric Field Mapping:

• Use equipotential surfaces (easier to measure)
• Conducting paper with voltage probes
• Map equal voltage points
• E-field perpendicular to equipotentials
• Field strength from potential gradient

Magnetic Field Mapping:

• Hall probe for quantitative measurements
• Compass array for direction visualization
• Iron filings for quick visualization
• Search coil for changing fields
• Calculate from measured forces

Data Analysis with E&M Specifics

E&M data often shows specific behaviors requiring targeted analysis:

Exponential Behaviors:

• Capacitor discharge: Q = Q₀e^(-t/RC)
• RL current growth: I = I₀(1 - e^(-t/τ))
• Linearize by logarithms
• Extract time constants from slopes

Sinusoidal Responses:

• AC circuit analysis
• Measure amplitude and phase
• Impedance from V/I ratio
• Phase from time delay: φ = ωΔt

Resonance Phenomena:

• LCR circuits show peaked response
• Measure amplitude vs. frequency
• Find resonance at maximum
• Q-factor from width: Q = f₀/Δf

Common Experimental Scenarios

Certain E&M measurements appear frequently with established procedures.

Resistivity Measurement

Multiple approaches for different materials:

Wire Resistivity:

• Measure resistance vs. length
• R = ρL/A predicts linear relationship
• Slope gives ρ/A
• Measure diameter for A
• Temperature dependence: ρ(T) = ρ₀(1 + αΔT)

Sheet Resistivity:

• Four-point probe method
• Eliminates contact resistance
• Current through outer probes
• Voltage across inner probes
• Geometry factors for finite samples

Semiconductor Resistivity:

• Van der Pauw method
• Temperature-dependent measurements
• Activation energy from Arrhenius plot
• Hall effect for carrier concentration

Capacitor Characterization

Beyond simple capacitance measurement:

Dielectric Constant:

• Parallel plate with changeable dielectric
• Measure C with and without material
• κ = C_with/C_without
• Control for edge effects
• Multiple materials for comparison

Voltage Dependence:

• Some capacitors vary with voltage
• Measure C at different bias voltages
• Plot C vs. V
• Extract voltage coefficients
• Important for varactors

Frequency Response:

• Real capacitors have parasitic inductance
• Impedance minimum at self-resonance
• Measure |Z| vs. frequency
• Model as RLC circuit
• Extract parasitic elements

Magnetic Field Measurements

Various techniques for different scenarios:

Helmholtz Coils:

• Create known uniform field
• Calibrate Hall probes
• B = (8/5√5)(μ₀NI/R) at center
• Map field uniformity
• Study superposition

Earth's Field:

• Tangent galvanometer method
• Compass deflection from known current
• B_earth = B_coil tan(θ)
• Multiple orientations for vector components
• Dip needle for inclination

Induced EMF Method:

• Rotating coil in field
• ε = NABω sin(ωt)
• Measure peak voltage
• Calculate B from known parameters
• Or flip coil for flux change

Detailed Rubric Breakdown

Understanding scoring maximizes credit even for imperfect procedures.

Part (a): Experimental Procedure (3-4 points)

Measurement Strategy (2 points):

• Clear identification of what to measure
• Realistic equipment choices
• Multiple measurements for reliability
• Controls for systematic errors

Procedural Clarity (1-2 points):

• Step-by-step instructions
• Diagrams of setup if helpful
• Safety considerations (high voltage?)
• Data recording plan

Common losses: Vague procedures, unrealistic equipment, missing multiple trials, safety hazards ignored.

Part (b): Analysis Plan (2-3 points)

Mathematical Framework (1-2 points):

• Correct physics relationships identified
• Linearization strategy explained
• Error propagation considered
• Graphical analysis planned

Practical Considerations (1 point):

• Realistic precision expectations
• Major error sources identified
• Strategies to minimize errors
• Validation checks described

Common losses: No linearization plan, ignoring measurement uncertainties, missing physics relationships.

Part (c): Data Analysis Execution (4-5 points)

Graph Quality (2 points):

• Appropriate quantities plotted
• Clear scales and labels with units
• All data points visible
• Error bars if appropriate

Parameter Extraction (2-3 points):

• Best-fit line drawn correctly
• Slope/intercept calculated
• Conversion to desired quantity shown
• Final answer with units and uncertainty

Common losses: Poor graph scales, calculation errors, missing unit conversions, unrealistic precision claims.

Advanced E&M Experimental Techniques

These approaches show sophisticated understanding.

Lock-In Detection Concepts

For weak signal measurement:

• Modulate signal at known frequency
• Multiply by reference signal
• Low-pass filter extracts DC component
• Rejects noise at other frequencies
• Mention for sensitive measurements

Bridge Circuits

Precision measurement technique:

• Wheatstone bridge for resistance
• Maxwell bridge for inductance
• Wien bridge for capacitance
• Null detection for high precision
• Eliminates meter limitations

Transmission Line Effects

For high-frequency measurements:

• Cable capacitance affects measurements
• Impedance matching important
• Reflections distort signals
• Probe compensation needed
• Relevant for fast pulses

Shielding and Grounding

Critical for clean measurements:

• Electrostatic shielding (Faraday cage)
• Magnetic shielding (mu-metal)
• Proper grounding prevents loops
• Twisted pairs reduce pickup
• Coaxial cables for high frequency

Time Management for E&M Experiments

With ~27 minutes total:

• Minutes 1-3: Understand measurement goal completely
• Minutes 4-8: Design procedure with equipment list
• Minutes 9-12: Explain analysis method clearly
• Minutes 13-15: Process given data (calculations)
• Minutes 16-22: Create high-quality graph
• Minutes 23-26: Extract results from graph
• Minutes 27: Quick reasonableness check

Prioritize the graph—it often carries the most points and demonstrates understanding even if calculations are incomplete.

E&M-Specific Pitfalls

Loading Effects Measurements affect circuits:

• Voltmeter has finite resistance
• Ammeter has small resistance
• Oscilloscope has input capacitance
• Choose meter ranges wisely

Ground Loops Multiple ground paths cause errors:

• Use single-point grounding
• Floating measurements when possible
• Differential measurements
• Isolation transformers

Frequency Limitations Real components have frequency limits:

• Resistors have parasitic inductance/capacitance
• Inductors have self-resonance
• Capacitors become inductive at high frequency
• Skin effect in conductors

Thermal Effects Temperature affects everything:

• Resistance temperature coefficients
• Thermoelectric voltages at junctions
• Component drift during measurement
• Self-heating from measurement current

Laboratory Reality Check

Know typical E&M measurement capabilities:

• Multimeters: ±0.1% for DC, worse for AC
• Oscilloscopes: 8-bit vertical, timing to ns
• Function generators: mHz to MHz typically
• Hall probes: mT resolution typically
• Electrometers: pA current, GΩ resistance

Reference realistic values to show experimental maturity.

Final Insights

E&M experimental design tests whether you can bridge the gap between Maxwell's equations and real measurements. The elegance of electromagnetic theory meets the messiness of actual circuits and fields.

Approach these problems practically: What can I actually measure? What affects my measurements? How do I extract what I want from what I can get? Your experimental design should feel like something you could actually do, with analysis sophisticated enough to extract meaningful results from imperfect data.

Remember that E&M measurements often involve inference—you measure voltage to find field, measure period to find inductance, measure phase to find reactance. The best experimental designs are simple in concept but clever in what they reveal. Let physics principles guide your design, and let mathematical analysis extract the maximum information from your measurements.