Mutual Interaction

AP Physics C: Topic 7.2

Topic: Forces Between Parallel Current-Carrying Conductors

The Double Rule

"To solve wire interactions, you must use RHR-1 to find the field created by the first wire, and then RHR-2 to find the force exerted on the second wire."

Interaction Logic

Wire 1 creates \(\vec{B}_1\)

\(\vec{B}_1\) exerts \(\vec{F}_2\) on Wire 2

I. The Mechanism of Attraction and Repulsion

One of the most profound experiments in classical electromagnetism involves two parallel wires carrying electric currents. We observe that they exert magnetic forces on each other, but the nature of the force—whether it is attractive or repulsive—depends entirely on the relative directions of the currents.

The Field-Force Link

In AP Physics C, we analyze this as a two-step process. Wire 1 generates a magnetic field in the surrounding space, including the location of Wire 2. Wire 2, being a conductor with moving charges, then experiences a Lorentz force due to that external field.

Fundamental Force Formula

\[ \frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi d} \]

The force per unit length (\(F/L\)) is proportional to the product of the currents and inversely proportional to the separation distance \(d\).

While the formula gives the magnitude, the direction is determined by applying the Right Hand Rules. If we fail to visualize the field orientation at the second wire, the direction of the force becomes impossible to predict.

II. Parallel Currents: Mutual Attraction

When current flows in the same direction in both wires, they experience a mutual attractive force. Let's break this down using our vector conventions.

Step 1: Wire 1 Field

Using RHR-1: Thumb up (\(I_1\)), fingers curl. At Wire 2's location, the field \(\vec{B}_1\) points INTO the page (\(\otimes\)).

Step 2: Force on Wire 2

Using RHR-2: Fingers up with \(I_2\), curl fingers IN with \(\vec{B}_1\). Thumb points LEFT, toward Wire 1.

Result: ATTRACTION

\(I_1\)

\(\vec{B}_1\) at Wire 2

Force (\(\vec{F}_{21}\))

\(I_2\)

Color Code: Field from W1 | Force on W2

A Note on Symmetry

By Newton's Third Law, if Wire 1 attracts Wire 2 to the left, Wire 2 must attract Wire 1 to the right with an equal and opposite force. The fields reinforce each other between the wires, but the resulting force pulls them together.

III. Anti-Parallel Currents: Mutual Repulsion

When currents flow in opposite directions, the magnetic interaction changes polarity, resulting in a force that pushes the conductors apart.

Step 1: Wire 1 Field

Using RHR-1: Thumb up (\(I_1\)), fingers curl. At Wire 2's location, the field \(\vec{B}_1\) points INTO the page (\(\otimes\)). (This step is identical to Case A).

Step 2: Force on Wire 2

Using RHR-2: Fingers point DOWN with \(I_2\), curl IN with \(\vec{B}_1\). Your thumb now points RIGHT, away from Wire 1.

Result: REPULSION

\(I_1\)

\(\vec{B}_1\) at Wire 2

Force (\(\vec{F}_{21}\))

\(I_2\)

Summary of Wire Rules

Like Currents (\(\uparrow\uparrow\)) \(\rightarrow\) Attract

Opposite Currents (\(\uparrow\downarrow\)) \(\rightarrow\) Repel

IV. Quantitative Mastery & SI Definitions

Calculating the Magnitude

To find the magnitude of the force per unit length, we combine the formula for the magnetic field of a long wire with the Lorentz force formula for a wire segment.

1. Magnetic Field from Wire 1: \[ B_1 = \frac{\mu_0 I_1}{2\pi d} \]

2. Force on Wire 2 segment: \[ F_{21} = I_2 L B_1 \sin(90^\circ) \]

3. Final Expression (Force per Length): \[ \frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi d} \]

The Official Definition of the Ampere

Interestingly, the SI unit for current—the **Ampere**—was historically defined based on this magnetic interaction between parallel wires rather than the flow of charge directly.

Historical SI Definition

"One Ampere is that constant current which, if maintained in two straight parallel conductors of infinite length and negligible circular cross-section, placed one meter apart in vacuum, would produce between these conductors a force equal to exactly \(2 \times 10^{-7}\) Newtons per meter of length."

Note: This definition implies \(\mu_0 = 4\pi \times 10^{-7}\) exactly. In 2019, the definition was updated to be based on the elementary charge \(e\), but the physical relationship remains a cornerstone of the field.

Mastery Check

• If you double both currents, the force quadruples (\(2 \times 2 = 4\)).
• If you double the distance, the force is halved.
• The force is always perpendicular to the wires.

© 2026 AP PHYSICS MASTERY Wire Interaction Blueprint ARTICLE-7.2-WIRE

Parallel Wire Showdown Article

Parallel Wire Showdown Article

Similar Materials

Truffula Tragedy Answer Key

Taryn Sigl

Truffula Tragedy Worksheet

Taryn Sigl

Teacher PD and Cheat Sheets Document

Vic Chamness

Future Architects Activity Sheet

Vic Chamness

Future Architects Presentation Slides

Vic Chamness

AI Inside Out Activity Sheet

Vic Chamness

AI Inside Out Presentation Slides

Vic Chamness

Pattern Detectives Activity Sheet

Vic Chamness

Pattern Detectives Presentation Slides

Vic Chamness

Smart Helpers Activity Worksheet

Vic Chamness

Smart Helpers Presentation Slides

Vic Chamness

Robot Brains Teacher Guide

Vic Chamness

Robot Brains Activity Worksheet

Vic Chamness

Robot Brains Presentation Slides

Vic Chamness

Blueprint Curriculum Overview Document

Vic Chamness

Survival Secrets Slides

Genetics Mastery Answer Key

Survival Secrets Answer Key

Survival Quest Quiz

Genetics Mastery Quiz

Inheritance Impact Worksheet

STAAR Checkpoint Assessment

Anole Taxonomy Trail Worksheet

Matter Master Answer Key

Protein Factory Worksheet

Matter Master Mission Slides

Mass Guardian Investigation Activity

Mission Control Answer Key

STAAR Practice Answer Key Force and Motion

Island Energy Architect Worksheet

Division Decisions Worksheet

STAAR Power Practice Force and Motion

STAAR Mission Assessment