FRQ 1 – Mathematical Routines

Overview

• Question 1 of 4 in Section II
• Worth 10 points (12.5% of your total exam score)
• Suggested time: 20-25 minutes
• Calculator allowed and essential
• Tests skills: Mathematical derivation with calculus (1.A, 1.C), Creating field/circuit diagrams (2.A, 2.B), Applying E&M principles (3.B, 3.C)

The Mathematical Routines question in Physics C: E&M demands sophisticated calculus-based analysis of electromagnetic phenomena. You'll calculate fields from continuous charge distributions using integration, derive potential functions, analyze time-varying circuits with differential equations, or find induced EMFs through Faraday's law. This question showcases calculus as the essential language for understanding electricity and magnetism beyond point charges and simple circuits.

Strategy Deep Dive

Mathematical Routines in E&M requires mastery of vector calculus and its physical interpretation. Success comes from recognizing which mathematical tools apply to which physical situations.

Vector Calculus Fundamentals

E&M inherently involves vector fields, making vector calculus essential:

• Dot products appear in flux calculations: Φ = ∫E⃗·dA⃗
• Cross products emerge in magnetic force: F⃗ = q(v⃗ × B⃗)
• Line integrals calculate work and EMF: ε = -∮E⃗·dl⃗
• Surface integrals determine flux through surfaces
• Gradients relate potential to field: E⃗ = -∇V

When setting up problems, immediately identify which vector operation applies. The physics dictates the mathematics.

Integration for Continuous Distributions

Unlike point charges, continuous distributions require integration:

Electric Field from Line Charge:

• Choose integration variable (often position along line)
• Express dq in terms of linear density: dq = λ dx
• Write dE⃗ from charge element: dE⃗ = k(dq/r²)r̂
• Identify components that survive integration
• Integrate, often using symmetry to eliminate components

Magnetic Field from Current Distributions:

• Use Biot-Savart: dB⃗ = (μ₀/4π)(I dl⃗ × r̂)/r²
• Express current element I dl⃗ appropriately
• Handle vector cross product before integrating
• Exploit symmetry to simplify

Key insight: Set up the integral thoughtfully. Good coordinate choice can transform impossible integrals into trivial ones.

Differential Equations in Circuits

Time-varying circuits lead to differential equations:

RC Circuits:

• Kirchhoff's loop rule: ε - q/C - IR = 0
• Since I = dq/dt: ε - q/C - R(dq/dt) = 0
• Rearrange: dq/dt + q/RC = ε/R
• Solve by separation or recognizing standard form

RL Circuits:

• Loop rule: ε - L(dI/dt) - IR = 0
• Rearrange: dI/dt + (R/L)I = ε/L
• Solution involves exponential approach to steady state

LC Oscillations:

• Energy conservation: ½LI² + q²/2C = constant
• Differentiate: LI(dI/dt) + (q/C)(dq/dt) = 0
• Since I = dq/dt: d²q/dt² + q/LC = 0
• Simple harmonic motion in charge!

Faraday's Law Applications

Induced EMF calculations showcase calculus directly:

ε = -dΦ/dt = -d/dt(∫B⃗·dA⃗)

Common scenarios:

• Changing B-field: Pull derivative inside integral
• Moving loop: Flux integral limits become time-dependent
• Rotating coil: Area projection varies as cos(ωt)
• Expanding loop: Both field and area may vary

Always identify what's changing with time before differentiating.

Common Problem Types

Certain scenarios dominate E&M Mathematical Routines, each with specific calculus techniques.

Gauss's Law with Symmetry

Problems with spherical, cylindrical, or planar symmetry:

Spherical:

• Charged sphere with radius R, total charge Q
• For r > R: E(4πr²) = Q/ε₀ → E = Q/4πε₀r²
• For r < R: Requires knowing charge distribution
• If uniform: E(4πr²) = Q(r³/R³)/ε₀

Cylindrical:

• Infinite line or cylinder of charge
• E(2πrL) = λL/ε₀ → E = λ/2πε₀r
• Direction radially outward from axis

Planar:

• Infinite sheet with surface charge σ
• E(2A) = σA/ε₀ → E = σ/2ε₀
• Field independent of distance!

The art lies in choosing the right Gaussian surface.

Ampere's Law Applications

For current distributions with symmetry:

Infinite Wire:

• ∮B⃗·dl⃗ = μ₀I_enclosed
• B(2πr) = μ₀I → B = μ₀I/2πr

Solenoid:

• Inside: B = μ₀nI (n = turns per length)
• Outside: B = 0 (ideal case)
• Demonstrates field confinement

Toroid:

• Similar to solenoid but curved
• Field confined to interior
• B varies with radius within toroid

Motional EMF Problems

Conducting rod moving through B-field:

Basic Setup:

• Rod length L, velocity v, field B
• Motional EMF: ε = BLv (if B⊥v⊥L)
• Derives from ε = ∮(v⃗×B⃗)·dl⃗

With Resistance:

• Induced current: I = BLv/R
• Magnetic force on rod: F = BIL = B²L²v/R
• Power dissipated: P = I²R = B²L²v²/R
• Mechanical power input: P = Fv = B²L²v²/R ✓

Energy conservation confirmed through mathematics!

Electromagnetic Induction Variations

Rotating Coil:

• Flux: Φ = NBA cos(ωt)
• EMF: ε = -dΦ/dt = NBAω sin(ωt)
• Peak EMF when coil parallel to field
• Basis for generators and motors

Mutual Inductance:

• Φ₂₁ = M₁₂I₁
• ε₂ = -M₁₂(dI₁/dt)
• Couples circuits electromagnetically
• Transformer principle

Detailed Rubric Breakdown

Understanding the scoring helps maximize partial credit.

Part (a): Setup and Field Representation (2-3 points)

Diagram Quality (1 point):

• Show field lines/equipotentials correctly
• Indicate current directions in circuits
• Label all components and coordinates
• Use vector notation properly

Initial Equations (1-2 points):

• State fundamental principle (Gauss's law, Faraday's law, etc.)
• Define coordinate system and variables
• Express charge/current elements correctly
• Show limits of integration

Common losses: Missing vector nature, incorrect limits, undefined variables.

Part (b): Calculus-Based Derivation (4-5 points)

Integration/Differentiation Setup (2 points):

• Correct mathematical expression of physical law
• Proper handling of vector components
• Appropriate substitution for variables
• Clear path to solution

Mathematical Execution (2 points):

• Perform calculus operations correctly
• Handle vector algebra properly
• Simplify expressions appropriately
• Arrive at correct functional form

Physical Consistency (1 point):

• Check dimensions throughout
• Verify expected limiting behavior
• Ensure signs reflect physics (attractive/repulsive)

Common losses: Vector algebra errors, missing constants, sign mistakes.

Part (c): Application and Analysis (3-4 points)

Numerical Calculation (1-2 points):

• Substitute values with units
• Calculate final result
• Express with appropriate precision
• Include proper units

Interpretation (1-2 points):

• Explain physical meaning of result
• Verify reasonableness
• Compare to known cases
• Predict behavior in limits

Common losses: Calculator errors, unreasonable results not caught, missing physics interpretation.

Advanced Techniques

These approaches demonstrate sophisticated understanding.

Exploiting Symmetry

Before integrating, always check for symmetry:

• Mirror symmetry → Components cancel
• Rotational symmetry → Use cylindrical/spherical coordinates
• Translational symmetry → Field independent of that coordinate

Example: Ring of charge on axis - only axial component survives by symmetry.

Dimensional Analysis Checks

Verify expressions using dimensions:

• Electric field: [Force]/[Charge] = N/C = V/m
• Magnetic field: [Force]/[Charge×Velocity] = T = Wb/m²
• Inductance: [Flux]/[Current] = H = Ω·s
• Capacitance: [Charge]/[Voltage] = F = C²·s²/kg·m²

Dimensional errors indicate conceptual mistakes.

Series Expansion Approximations

For complex expressions, Taylor series helps:

• Far from dipole: E ≈ kp/r³ (where p = qd)
• Small angle: sin θ ≈ θ, cos θ ≈ 1 - θ²/2
• Weak damping: e^(-γt) ≈ 1 - γt

Shows physical insight into dominant behaviors.

Time Management

With ~22 minutes for Mathematical Routines:

• Minutes 1-3: Parse problem, identify key physics principle
• Minutes 4-6: Draw diagrams, set up coordinates
• Minutes 7-14: Core derivation with careful calculus
• Minutes 15-19: Apply to specific case, calculate
• Minutes 20-22: Verify reasonableness, check units

If behind, show clear setup even if integration incomplete. Partial credit rewards understanding the approach.

Calculator Strategies for E&M

Use your calculator effectively:

• Store physical constants (ε₀, μ₀, e, etc.)
• Check integral results numerically
• Graph potential/field functions
• Verify exponential time constants

But always show analytical work first!

Common Pitfalls

Vector Confusion E&M is inherently three-dimensional. Common errors:

• Forgetting vector nature of fields
• Incorrect dot/cross products
• Missing component analysis
• Wrong direction assignments

Always draw vectors and components clearly.

Integration Limits Carefully consider integration bounds:

• Charge distributions: physical extent
• Flux integrals: closed surface
• Line integrals: complete path
• Time integrals: initial to final state

Sign Conventions Signs carry physical meaning:

• Negative charge → field points inward
• Lenz's law → negative sign in Faraday's law
• Potential decreases in field direction
• Current direction affects magnetic field via right-hand rule

Final Insights

E&M Mathematical Routines showcase the power of calculus in physics. Maxwell's equations, written properly, are differential equations. The discrete becomes continuous, the finite becomes infinitesimal, and calculus bridges the gap.

Approach these problems systematically: What's the charge/current distribution? What field quantity do I need? What symmetry can I exploit? What calculus tool applies? Let the physics guide the mathematics, and let the mathematics reveal the physics.

Remember, you're not just calculating—you're uncovering how electromagnetic fields behave in space and time. Every integral represents summing infinitesimal contributions, every derivative captures instantaneous change. Master these connections, and E&M transforms from abstract mathematics to tangible understanding of the electromagnetic world.