SECTION 1
Historical Context & Motivation
For centuries, electricity and magnetism were thought to be completely separate phenomena with no connection to each other. People understood lodestones as natural magnets and static electricity as a parlor trick, but nobody suspected a deeper link. That all changed in 1820, when a Danish professor made a startling observation during a lecture demonstration. His discovery launched an entirely new branch of physics called electromagnetism, which unified two forces that had seemed unrelated. The race to understand the connection between electric current and magnetic fields would transform science and engineering within a single generation.
1820 • Ørsted's Discovery
1820 • Ampère's Mathematical Law
1831 • Faraday's Electromagnetic Induction
1865 • Maxwell's Equations
The central question driving this lesson is straightforward: how exactly does an electric current create a magnetic field, and what determines that field's strength and direction? Understanding this relationship is the foundation for motors, generators, transformers, and nearly every electrical device you use. By the end of this lesson, you will be able to predict the magnetic field around a current-carrying wire using both conceptual reasoning and mathematical tools.
SECTION 2
Core Principles & Definitions
Before diving into calculations, you need to understand a few foundational ideas that connect electric current to magnetism. These principles form the conceptual backbone of everything that follows. Each one builds on the previous, so take them in order.
Moving Charges Create Magnetic Fields
The Right-Hand Rule
Field Strength Depends on Current and Distance
Superposition of Magnetic Fields
SECTION 3
Visual Explanation: The Magnetic Field Around a Wire
The diagram below shows a long, straight wire carrying conventional current upward (out of the page). The magnetic field lines form concentric circles centered on the wire. Notice how the circles are more closely spaced near the wire and spread apart farther away, indicating that the field strength decreases with distance.
There are two important symbols to remember when viewing cross-sectional diagrams of wires. A dot (⊙) means the current flows out of the page toward you, like the tip of an arrow coming at you. An X (⊗) means the current flows into the page away from you, like the tail feathers of an arrow flying away. These conventions let us represent three-dimensional current directions in a flat diagram.
SECTION 4
Mathematical Framework
Now that you understand the conceptual picture, it is time to put numbers to it. The key equation for the magnetic field around a long, straight wire was developed from Ampère's law. This formula lets you calculate the exact magnetic field strength at any distance from a current-carrying wire.
MAGNETIC FIELD OF A LONG STRAIGHT WIRE
B = magnetic field strength (in tesla, T); μ₀ = permeability of free space = 4π × 10⁻⁷ T·m/A; I = current in the wire (in amperes, A); r = perpendicular distance from the wire (in meters, m).
This equation tells us two important proportionalities. First, B is directly proportional to I — doubling the current doubles the magnetic field. Second, B is inversely proportional to r — doubling the distance cuts the field in half. The constant μ₀ is a fundamental property of empty space that describes how easily magnetic fields can form in a vacuum.
MAGNETIC FIELD AT CENTER OF A SOLENOID
n = number of turns per unit length (turns/m); I = current (A). A solenoid is a coil of wire that creates a nearly uniform field inside, behaving like a bar magnet.
FORCE BETWEEN TWO PARALLEL WIRES
F/L = force per unit length (N/m); I₁ and I₂ = currents in each wire (A); d = distance between wires (m). Parallel currents (same direction) attract; antiparallel currents (opposite directions) repel.
SECTION 5
Magnetic Fields from Two Parallel Wires
When two parallel wires each carry current, the magnetic field at any point is the vector sum of the individual fields from each wire. The behavior at the midpoint between the wires depends critically on whether the currents flow in the same direction (parallel) or in opposite directions (antiparallel). This distinction is essential and a common source of confusion, so study the diagram carefully.
The key insight here requires careful application of the right-hand rule at the midpoint for each wire individually. For the parallel case (both currents out of page), consider the midpoint between them. Wire 1's field at the midpoint — which is to the right of wire 1 — points upward. Wire 2's field at the midpoint — which is to the left of wire 2 — points downward. Since the two field contributions are in opposite directions, they cancel. For the antiparallel case, wire 1 (out of page) still produces an upward field at the midpoint, but wire 2 (into the page) also produces an upward field at the midpoint (to its left). Both contributions are in the same direction, so they add.
SECTION 6
Worked Example
Let's work through a complete problem that brings together the equation for a single wire's magnetic field and the concept of superposition for two wires.
Worked example
- 1
Step 1 — Read the Problem
Two long, straight, parallel wires are separated by a distance d = 0.20 m. Wire 1 carries a current I₁ = 5.0 A to the north, and wire 2 carries a current I₂ = 3.0 A to the south. Find the magnitude and direction of the net magnetic field at the midpoint between the two wires. - 2
Step 2 — Identify Given Values
d = 0.20 m, so the midpoint is at r = 0.10 m from each wire. I₁ = 5.0 A (northward). I₂ = 3.0 A (southward). μ₀ = 4π × 10⁻⁷ T·m/A. Because the currents are antiparallel, the fields at the midpoint will point in the same direction and add. - 3
Step 3 — Determine Field Directions Using the Right-Hand Rule
For wire 1 (current north): point your right thumb north. At the midpoint (which is to the east of wire 1 if we view from above with wire 1 on the left), your fingers curl into the page. So B₁ points into the page at the midpoint. For wire 2 (current south): point your right thumb south. At the midpoint (which is to the west of wire 2), your fingers also curl into the page. So B₂ also points into the page. Since both fields point into the page, they add. - 4
Step 4 — Calculate Individual Field Magnitudes
B₁ = μ₀I₁ / (2πr) = (4π × 10⁻⁷)(5.0) / (2π × 0.10). Simplify: the π cancels, giving B₁ = (4 × 10⁻⁷ × 5.0) / (2 × 0.10) = (2.0 × 10⁻⁶) / (0.20) = 1.0 × 10⁻⁵ T. Similarly, B₂ = (4π × 10⁻⁷)(3.0) / (2π × 0.10) = (4 × 10⁻⁷ × 3.0) / (0.20) = (1.2 × 10⁻⁶) / (0.20) = 6.0 × 10⁻⁶ T.B₁ = 1.0 × 10⁻⁵ T; B₂ = 6.0 × 10⁻⁶ T - 5
Step 5 — Find the Net Field
Because both fields point into the page, we add their magnitudes: B_net = B₁ + B₂ = 1.0 × 10⁻⁵ + 6.0 × 10⁻⁶ = 1.6 × 10⁻⁵ T. The net magnetic field at the midpoint is 1.6 × 10⁻⁵ T directed into the page.B_net = 1.6 × 10⁻⁵ T, directed into the page
SECTION 7
Applications and Limitations
The relationship between electric current and magnetic fields is the basis for an enormous range of technologies. However, the simple equations we use in this lesson have important limitations. The table below compares key real-world applications with the assumptions and limits of our mathematical models.
| Application / Feature | How It Uses Current–Field Relationship | Model Limitations |
|---|---|---|
| Electric motors | Current through a coil in a magnetic field creates a torque that spins a rotor, converting electrical energy to mechanical energy. | Our straight-wire formula does not account for coil geometry; solenoid and loop formulas are needed. |
| Electromagnets (MRI, scrapyard) | A solenoid with many turns produces a strong, nearly uniform field inside, intensified by an iron core. | Our model assumes vacuum; ferromagnetic cores multiply the field by factors of 100–10,000, requiring additional material constants. |
| Generators | A spinning coil in a magnetic field induces a changing magnetic flux, which creates an alternating current (Faraday's law). | Our lesson focuses on static fields from steady currents; generators require time-varying field analysis. |
| Transformers | Alternating current in a primary coil creates a changing magnetic field that induces a voltage in a secondary coil, allowing voltage conversion. | Requires AC, not DC; our equations apply to steady (DC) currents and do not cover electromagnetic induction. |
Real-world applications and the limits of the straight-wire/solenoid models
SECTION 8
Connections to Advanced Electromagnetism
The ideas in this lesson form the starting point for a much richer theory. In advanced physics courses, you will encounter more powerful mathematical tools that generalize what we have learned. The table below shows how the concepts from this lesson connect to their more advanced counterparts.
| This Lesson (Introductory Level) | Advanced Theory |
|---|---|
| B = μ₀I/(2πr) for a long straight wire | Biot–Savart law: dB = (μ₀/4π) × (Idl × r̂)/r², which calculates the field from any shape of current-carrying conductor. |
| Right-hand rule for field direction | Cross-product vector mathematics (v × B) used systematically in three dimensions. |
| Superposition: add fields from two wires | Ampère's law in integral form: ∮B·dl = μ₀I_enclosed, which exploits symmetry to find fields in complex geometries. |
| Steady currents produce steady magnetic fields | Maxwell's equations unify changing electric and magnetic fields, predicting electromagnetic waves and describing all classical electromagnetism. |
Introductory vs. advanced electromagnetism concepts
If you continue to AP Physics C or a college-level electricity and magnetism course, you will learn the Biot–Savart law and Ampère's law in their full vector-calculus forms. These tools let you calculate the magnetic field from any arbitrary current distribution, not just straight wires and solenoids. Maxwell's equations — the crown jewel of classical physics — show that changing electric fields also create magnetic fields, completing the symmetry that Faraday first glimpsed in 1831.
SECTION 9
Practice Problems
Question 1: CONCEPTUAL
- A. The needle deflects toward geographic north.
- B. The needle deflects toward geographic south.
- C. The needle deflects straight upward, away from the wire.
- D. The needle aligns with the current direction, pointing east.
Show answer
Question 2: BASIC CALCULATION
- A. 8.0 × 10−6 T
- B. 4.0 × 10−6 T
- C. 1.6 × 10−5 T
- D. 2.0 × 10−6 T
Show answer
Question 3: INTERMEDIATE
- A. The net magnetic field at P is zero; the wires attract each other.
- B. The net magnetic field at P is doubled (fields add); the wires attract each other.
- C. The net magnetic field at P is zero; the wires repel each other.
- D. The net magnetic field at P is doubled (fields add); the wires repel each other.
Show answer
Question 4: APPLIED
- A. Double the current to 6.0 A; new field is 2.4 × 10−5 T
- B. Quadruple the current to 12.0 A; new field is 4.8 × 10−5 T
- C. Double the current to 6.0 A; new field is 1.2 × 10−5 T
- D. Increase current by √2 to about 4.2 A; new field is 2.4 × 10−5 T
Show answer
Question 5: CRITICAL THINKING
- A. The fields at P add, and the wires repel each other.
- B. The fields at P cancel, and the wires attract each other.
- C. The fields at P cancel, and the wires repel each other.
- D. The fields at P add, and the wires attract each other.