AP Physics C E and M How to Solve: Biot-Savart Law & Ampère’s Law (Straight Wire, Loop, Solenoid)

How to Solve: Biot-Savart Law & Ampère’s Law (Straight Wire, Loop, Solenoid)

Introduction

Mastering Biot-Savart and Ampère’s Law lets you predict magnetic fields from any current-carrying wire—a 10-15% question on the AP Physics C: E&M exam. Miss this, and you lose easy points on free-response problems about motors, MRI machines, or even Earth’s magnetic field.

WHAT YOU NEED TO KNOW FIRST

1. Right-Hand Rule (RHR) for currents: Thumb = current direction, fingers = magnetic field direction.
2. Vector cross-product basics: Direction of d? × r̂ (Biot-Savart) or I × d? (Ampère’s).
3. Symmetry in magnetic fields: Straight wires, loops, and solenoids have predictable field patterns.

KEY TERMS & FORMULAS

1. Biot-Savart Law (General Case)

Formula: dB = (μ₀ / 4π) (I d? × r̂) / r² - dB: Magnetic field contribution from a tiny wire segment (T) - μ₀: Permeability of free space (MEMORISE THIS: 4π × 10⁻⁷ T·m/A) - I: Current (A) - d?: Tiny wire segment (vector, direction = current flow) - r̂: Unit vector pointing from wire segment to field point - r: Distance from wire segment to field point (m)

When to use: When the wire isn’t straight, a loop, or a solenoid (e.g., bent wires, arcs).

2. Ampère’s Law (Symmetry Cases)

Formula: ∮B·d? = μ₀ I_enc - ∮B·d?: Line integral of magnetic field around a closed loop (T·m) - I_enc: Current enclosed by the loop (A)

When to use: Only for high-symmetry cases (straight wires, loops, solenoids, toroids).

3. Special Cases (Given on AP Exam Sheet)

A. Long Straight Wire

Formula: B = (μ₀ I) / (2π r) - B: Magnetic field (T) - r: Perpendicular distance from wire (m)

Direction: Use RHR—thumb = current, fingers curl in B direction.

B. Circular Loop (Center)

Formula: B = (μ₀ I) / (2 R) - R: Radius of loop (m)

Direction: RHR—curl fingers in current direction, thumb = B direction.

C. Solenoid (Inside, Long)

Formula: B = μ₀ n I - n: Turns per unit length (turns/m)

Direction: RHR—curl fingers in current direction, thumb = B direction.

STEP-BY-STEP METHOD

For Biot-Savart Law (General Case)

1. Draw the wire and field point: Label current direction (I) and the point where you need B.
2. Pick a tiny segment d?: Choose a small piece of wire where current flows.
3. Find r and r̂:
4. r: Distance from d? to field point.
5. r̂: Unit vector from d? to field point (draw an arrow).
6. Compute d? × r̂:
7. Use RHR: Point fingers in d? direction, curl toward r̂. Thumb = dB direction.
8. Write dB expression:
9. dB = (μ₀ / 4π) (I d? sinθ) / r²
10. θ = angle between d? and r̂ (usually 90° for straight wires).
11. Integrate over the wire: Add up all dB contributions (often simplifies to a single term for symmetric cases).

For Ampère’s Law (Symmetry Cases)

1. Check symmetry: Is the field constant along a path? (Straight wire, loop, solenoid.)
2. Draw Amperian loop:
3. Straight wire: Circle around wire (radius r).
4. Solenoid: Rectangle (one side inside, one outside).
5. Compute ∮B·d?:
6. For straight wire: B 2πr (B is constant along the circle).
7. For solenoid: B L (L = length of inner side).
8. Find I_enc: Current passing through the loop.
9. Set equal to μ₀ I_enc: Solve for B.

WORKED EXAMPLES

Example 1 – Basic: Straight Wire (Ampère’s Law)

Problem: A long straight wire carries 5 A. What is B at 0.1 m from the wire?

Steps:
1. Symmetry check: Straight wire → use Ampère’s Law.
2. Amperian loop: Circle of radius 0.1 m around wire.
3. ∮B·d? = B 2πr = B 2π(0.1)
4. I_enc = 5 A
5. Set equal: B 2π(0.1) = μ₀ 5
6. Solve for B: B = (μ₀ 5) / (2π 0.1) = (4π × 10⁻⁷ 5) / (0.2π) = 1 × 10⁻⁵ T

What we did and why: We used Ampère’s Law because the wire is straight (high symmetry). The Amperian loop was a circle, so the integral simplified to B circumference. We plugged in the enclosed current and solved for B.

Example 2 – Medium: Circular Loop (Biot-Savart)

Problem: A circular loop of radius 0.2 m carries 3 A. Find B at the center.

Steps:
1. Draw loop and field point: Center of loop.
2. Pick d?: Tiny segment of wire.
3. r = R = 0.2 m, r̂ points from d? to center.
4. d? × r̂: Always 90° (sinθ = 1).
5. dB = (μ₀ / 4π) (I d?) / R²
6. Integrate over loop: ∫d? = 2πR (full circumference).
7. B = (μ₀ / 4π) (I 2πR) / R² = (μ₀ I) / (2 R)
8. Plug in numbers: B = (4π × 10⁻⁷ 3) / (2 0.2) = 9.42 × 10⁻⁶ T

What we did and why: We used Biot-Savart because the loop isn’t a straight wire. The integral simplified because all d? segments contribute equally to B at the center. The final formula matches the "given" loop formula, confirming our work.

Example 3 – Exam-Style: Solenoid (Ampère’s Law)

Problem: A solenoid has 200 turns over 0.5 m and carries 2 A. What is B inside?

Steps:
1. Symmetry check: Solenoid → use Ampère’s Law.
2. Amperian loop: Rectangle (length L = 0.1 m inside, negligible outside).
3. ∮B·d? = B L (B is constant inside, zero outside).
4. n = 200 turns / 0.5 m = 400 turns/m
5. I_enc = n L I = 400 0.1 2 = 80 A
6. Set equal: B 0.1 = μ₀ 80
7. Solve for B: B = (4π × 10⁻⁷ 80) / 0.1 = 1.0 × 10⁻³ T

What we did and why: We used Ampère’s Law because the solenoid has high symmetry. The Amperian loop was a rectangle, and only the inner segment contributed to the integral. We calculated the enclosed current using turns per unit length.

COMMON MISTAKES

1. MISTAKE: Forgetting μ₀ (4π × 10⁻⁷). WHY IT HAPPENS: Students confuse it with ε₀ or omit it entirely. CORRECT APPROACH: MEMORISE μ₀ = 4π × 10⁻⁷ T·m/A. Write it at the top of your page.

2. MISTAKE: Mixing up r and R in loop problems. WHY IT HAPPENS: Using the wrong distance (e.g., radius vs. distance from wire). CORRECT APPROACH: Label your diagram: R = loop radius, r = distance from wire.

3. MISTAKE: Wrong direction for B (RHR errors). WHY IT HAPPENS: Thumb/fingers misaligned. CORRECT APPROACH: Practice RHR daily: Thumb = current, fingers = B direction.

4. MISTAKE: Using Biot-Savart for straight wires. WHY IT HAPPENS: Overcomplicating symmetric cases. CORRECT APPROACH: Use Ampère’s Law for straight wires, loops, solenoids. Biot-Savart is for irregular shapes.

5. MISTAKE: Ignoring units (e.g., cm vs. m). WHY IT HAPPENS: Carelessness with conversions. CORRECT APPROACH: Convert all distances to meters before plugging in.

EXAM TRAPS

1. TRAP: "Long wire" vs. "finite wire" in problems. HOW TO SPOT IT: If the problem says "long wire," use B = μ₀I / (2πr). If it’s finite, you must use Biot-Savart. HOW TO AVOID IT: Underline "long" or "finite" in the problem. If unsure, assume "long" for AP exams.

2. TRAP: Multiple currents in Ampère’s Law. HOW TO SPOT IT: A problem with two wires or a loop with multiple turns. HOW TO AVOID IT: Sum all enclosed currents (I_enc = I₁ + I₂ + ...). Watch for direction (use RHR to check signs).

3. TRAP: Solenoid with non-uniform turns. HOW TO SPOT IT: Problem mentions "turns per unit length" but gives total turns and length. HOW TO AVOID IT: Calculate n = N / L first. Never plug in total turns directly.

1-MINUTE RECAP

"Listen up—this is your 60-second crash course for Biot-Savart and Ampère’s Law. First, memorise μ₀ = 4π × 10⁻⁷. For straight wires, loops, or solenoids, use Ampère’s Law—draw a loop, compute ∮B·d?, set equal to μ₀I_enc, and solve for B. For weird shapes, use Biot-Savart: pick a d?, find r and r̂, compute d? × r̂, and integrate. Always check symmetry—if it’s symmetric, Ampère’s is faster. RHR is your best friend: thumb = current, fingers = B direction. Watch for units (meters!) and enclosed current signs. Now go crush that exam!"