AP Physics C: E&M Practice 2: Mathematical Routines is the science practice where you do the math of physics. You derive symbolic expressions, calculate or estimate unknown quantities with units, compare quantities across scenarios, and predict how values change when variables change.

This practice is the heaviest-weighted one on the exam. On the multiple-choice section, Practice 2 skills are worth roughly 25 to 30 percent (2.A), 20 to 25 percent (2.B), and 10 to 15 percent each for 2.C and 2.D. On the free-response section, Practice 2 carries about 40 to 45 percent of the weighting. If you get comfortable with these four moves, you cover a large chunk of the exam.

This guide breaks down each subskill, shows how they appear across all six units, and gives you targeted practice ideas.

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What Practice 2: Mathematical Routines Means

The grouping description is short and direct: conduct analyses to derive, calculate, estimate, or predict. Each verb maps to one subskill.

Derive (2.A): Build a symbolic expression from known quantities by choosing a logical math pathway.
Calculate or estimate (2.B): Find an unknown number with correct units by following a logical computational pathway.
Compare (2.C): Relate physical quantities between two or more scenarios, or between different times or locations in one scenario.
Predict (2.D): Use functional dependence between variables to find new values or factors of change.

The common thread is choosing a pathway. You are not just plugging into one formula. You select the right principle, connect equations, and follow the chain to an answer.

What This Practice Requires

Here is what each subskill asks you to do in practice.

2.A Derive a symbolic expression

Start from a fundamental principle or a reference equation.
Combine relationships to reach an expression in the requested variables.
Keep the answer symbolic. No numbers plugged in unless asked.

2.B Calculate or estimate with units

Identify known quantities and the target unknown.
Pick the equations that connect them.
Carry units through the whole calculation and report the answer with correct units.

2.C Compare quantities

Set up the relevant relationship for each scenario or location.
Decide which quantity is larger, smaller, or equal, and why.
Often you reason proportionally rather than computing exact numbers.

2.D Predict new values or factors of change

Identify how the target quantity depends on each variable.
Apply that functional dependence when a variable changes by a known factor.
Report the new value or the factor of change.

Skills You Need for This Practice

You will use the same math toolkit across all four subskills.

Calculus. This is a calculus-based course. Expect integrals for charge distributions, flux, and work, plus derivatives for current as the rate of change of charge or for induced emf.
Algebra and proportional reasoning. Solving systems, isolating variables, and recognizing how one quantity scales with another.
Vector reasoning. Adding fields and forces with direction, using components, and applying right-hand rules.
Unit tracking. Reporting answers in correct SI units and using units to check whether an answer makes sense.
Reference equations. Knowing which fundamental principle to start from. On the FRQ, you are often asked to begin a derivation by writing a fundamental physics principle or an equation from the reference information.

How It Shows Up on the AP Exam

Multiple-choice

2.A and 2.B together dominate the calculation-style questions. Expect to derive a symbolic answer choice (2.A) or compute a number with units (2.B).
2.C and 2.D show up as comparison and scaling questions. These often have no arithmetic and reward proportional reasoning.
A four-function, scientific, or graphing calculator is allowed on both sections.

Free-response

Practice 2 is heavily represented, roughly 40 to 45 percent of the FR weighting.
FRQ 1 is the Mathematical Routines question type, which shares a name with this science practice. Practice 2 skills also appear across the other free-response question types.
Derivation prompts usually say to begin with a fundamental principle or reference equation, then show each step to the requested expression.

These weightings come from the course framework. Treat any solving strategies below as practical advice, not official scoring rules.

Examples Across the Course

These examples are drawn from sample questions across different units so you can see each subskill in context.

2.A Derive, Unit 8 (Gauss's Law) A cube of side length $a$ sits with one corner at the origin in a field $\vec{E} = (bx)\hat{i}$ . Find the total flux. The field only varies along $x$ , so flux through the $x = a$ face is $E A = (ba)(a^2)$ and the $x = 0$ face contributes zero. Net flux is $ba^3$ . You derived the result by selecting the flux definition and following it through.

2.B Calculate with units, Unit 11 (Resistivity) A wire of length 0.25 m and diameter $3.0 \times 10^{-3}$ m dissipates $6.0 \times 10^{-4}$ W with current 0.50 A. Find resistivity. Use $P = I^2 R$ to get $R$ , then $R = \rho L / A$ with $A = \pi r^2$ to solve for $\rho$ . The pathway gives about $6.8 \times 10^{-8}\ \Omega \cdot \text{m}$ , with units carried throughout.

2.B Calculate, Unit 9 (Electric Potential) A proton moves from A to B in a uniform 1000 V/m field along $+y$ , with the A-to-B line at $37^{\circ}$ to the $x$ -axis over 5 m. The change in potential energy depends on the displacement along the field. The vertical displacement is $5\sin 37^{\circ} = 3$ m, giving $\Delta U = -3000$ eV for the positive charge moving against the field direction.

2.C Compare, Unit 8 (Electric Fields) Two charges $-3Q$ and $+4Q$ sit on equally spaced tick marks. Find where the net field magnitude is greatest. You compare $1/r^2$ contributions and their directions at each labeled point rather than computing exact field values. The point closest to the larger charge with reinforcing directions wins.

2.D Predict, Unit 11 (Internal Resistance) A nonideal battery gives current $I$ through resistance $R$ and current $I/2$ through $3R$ . Using $\mathcal{E} = I(R + r)$ for each case, set the two emf expressions equal and solve. The functional dependence between current and total resistance gives $r = R$ .

2.A Derive, Unit 12 (Magnetic Fields of Wires) Three parallel wires contribute fields at the origin. Add each $B = \mu_0 I / (2\pi d)$ contribution as a vector using the right-hand rule. Summing magnitudes and directions yields $2\mu_0 I / (\pi d)$ in the $-z$ -direction.

How to Practice Practice 2: Mathematical Routines

Try these strategies. They are practical study advice, not official rules.

Always name your starting principle. For derivations, write the fundamental equation first. This earns credit and keeps your pathway organized.
Solve symbolically, then plug in. Keep variables until the last step so you can check dimensions and reuse the result.
Track units in every line. If your final units are wrong, the pathway has an error.
Practice proportional reasoning out loud. For 2.C and 2.D, ask how the quantity depends on each variable, then state the factor of change without computing.
Redo MCQs as derivations. Take a calculation question and write the full symbolic derivation. This builds 2.A and 2.B at the same time.
Mix units when you practice. Pull problems from circuits, fields, flux, and induction in one session so you practice choosing the right pathway, not just repeating one formula.

Common Mistakes

Plugging in numbers too early. This makes derivation errors hard to catch and can cost credit on symbolic answers.
Dropping units. Many MCQ distractors are off by a power of ten or have wrong units. Tracking units catches these.
Ignoring direction. Field and force problems require vector addition, not just magnitude. Use components and right-hand rules.
Treating comparisons as full calculations. For 2.C and 2.D, reason proportionally. Computing exact numbers wastes time and invites arithmetic errors.
Forgetting functional dependence. If a quantity goes as $1/r^2$ and $r$ doubles, the quantity drops by a factor of 4, not 2.
Skipping the starting principle on FRQs. Derivation prompts often require you to begin from a reference equation. Skipping this loses points.

Quick Review

Practice 2 = derive, calculate or estimate, compare, predict. It is the highest-weighted practice on the exam.
2.A: Build symbolic expressions from a logical math pathway.
2.B: Compute unknowns with correct units from a logical computational pathway.
2.C: Compare quantities across scenarios, times, or locations, usually with proportional reasoning.
2.D: Predict new values or factors of change using functional dependence.
MCQ weighting: about 25 to 30 percent (2.A), 20 to 25 percent (2.B), 10 to 15 percent each (2.C, 2.D).
FR weighting: about 40 to 45 percent across the practice.
Habits that help: start from a named principle, stay symbolic, carry units, and reason proportionally for comparisons.