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

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

Force per unit length depends on \(\mu_0\), currents \(I_1, I_2\), and 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 (\(\vec{F}_{21}\))

\(I_2\)

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 (\(\otimes\)).

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

\(I_2\)

IV. Mapping the Fields: Individual Contributions

To truly understand why the force occurs, we must examine the magnetic field geometry created by each wire independently. Every wire acts as a source of circular field loops. When two wires are near each other, their fields overlap at every point in space.

Magnetic Field of Wire 1 (\(\vec{B}_1\))

OUT (\(\odot\))

IN (\(\otimes\))

Wire 1 creates field loops that are "out" on its left and "in" on its right.

Magnetic Field of Wire 2 (\(\vec{B}_2\))

OUT (\(\odot\))

IN (\(\otimes\))

Wire 2 also creates its own field loops, independent of Wire 1's field.

The Region Between the Wires

In the space directly between the two wires, the fields from Wire 1 and Wire 2 interact. This is the key to understanding the net interaction:

Case 1: Parallel Currents

Between the wires, \(\vec{B}_1\) is IN (\(\otimes\)) and \(\vec{B}_2\) is OUT (\(\odot\)). These fields point in opposite directions and partially cancel each other out. This creates a region of low field density, and the wires are "pushed" into it from the high-density outer regions.

Case 2: Anti-Parallel Currents

Between the wires, if \(I_2\) is reversed, its field \(\vec{B}_2\) is now IN (\(\otimes\)). Since \(\vec{B}_1\) is also IN (\(\otimes\)), they point in the same direction and reinforce each other. This creates a region of high field density, and the wires are "pushed" away from each other.

V. 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

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."

Mastery Check

• Individual Fields: Each wire creates its own B-field regardless of the other.
• Cancellation: Same-direction currents have fields that cancel between the wires \(\rightarrow\) Attraction.
• Reinforcement: Opposite-direction currents have fields that reinforce between the wires \(\rightarrow\) Repulsion.

© 2026 AP PHYSICS MASTERY Wire Interaction Blueprint Page 05 of 05

Parallel Wire Showdown Article Revised

Parallel Wire Showdown Article Revised

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