Why do parallel currents attract or repel each other?

This phenomenon arises from two fundamental electromagnetic principles. First, every current-carrying wire generates a magnetic field around itself. Second, any current-carrying wire placed within an external magnetic field experiences a force. So, when two parallel wires carry currents, each wire produces a magnetic field that then exerts a force on the other wire. The direction of this force (attraction or repulsion) depends on how these magnetic fields interact, which in turn depends on the relative directions of the currents. Same-direction currents lead to attraction, while opposite-direction currents lead to repulsion.

How is the direction of the force determined?

The direction of the force is determined by a two-step process involving standard right-hand rules. First, use the Right-Hand Thumb Rule to find the direction of the magnetic field produced by one wire at the location of the second wire. Point your thumb in the direction of the current in the first wire; your curled fingers indicate the magnetic field lines. Second, use Fleming's Left-Hand Rule (or the vector cross product $\vec{F} = I(\vec{L} \times \vec{B})$) to find the force on the second wire. Point your forefinger in the direction of the magnetic field, your middle finger in the direction of the current in the second wire, and your thumb will indicate the direction of the force.

How is the Ampere defined using the force between parallel currents?

The Ampere, the SI unit of electric current, is precisely defined based on this phenomenon. One Ampere is defined as the constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed one metre apart in vacuum, would produce between these conductors a force exactly equal to $2 \times 10^{-7}$ Newton per metre of length. This definition provides a practical and reproducible standard for the unit of current.

What happens if there are three parallel wires instead of two?

If there are three or more parallel wires, the net force on any one wire is the vector sum of the forces exerted on it by each of the other wires individually. For example, if you want to find the force on wire A, you would calculate the force on A due to B, and the force on A due to C. Then, you would add these two forces vectorially, considering their directions (attraction or repulsion), to find the total force on wire A. This requires careful application of the direction rules for each pair of interacting wires.

Force between Parallel Currents

Q: What is the formula for the force between two parallel currents?

The magnitude of the force per unit length ($F/L$) between two long, straight, parallel current-carrying conductors separated by a distance $d$ is given by the formula: $$\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi d}$$ where $\mu_0$ is the permeability of free space ($4\pi \times 10^{-7}\,\text{T}\cdot\text{m/A}$), $I_1$ and $I_2$ are the magnitudes of the currents in the two wires, and $d$ is the perpendicular distance between them. This formula is crucial for both conceptual understanding and numerical problem-solving.

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Version 1Updated 24 Mar 2026

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The fundamental principle governing the interaction between parallel current-carrying conductors states that two such conductors exert a force on each other due to the magnetic fields they produce. Each current creates a magnetic field in the surrounding space, and this magnetic field then exerts a Lorentz force on the other current-carrying conductor. The direction of this force is determined by …

Quick Summary

The interaction between parallel current-carrying wires is a fundamental concept in electromagnetism. Each wire carrying a current generates a magnetic field around it. When a second current-carrying wire is placed within this magnetic field, it experiences a force.

This force's direction depends on the relative directions of the currents: parallel currents flowing in the same direction attract each other, while those flowing in opposite directions repel. The magnitude of this force per unit length is directly proportional to the product of the currents and inversely proportional to their separation distance.

The formula for this force per unit length is $\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi d}$ . This principle is not only crucial for understanding electromagnetic interactions but also forms the basis for the precise definition of the Ampere, the SI unit of electric current.

Correctly applying the Right-Hand Thumb Rule for magnetic field direction and Fleming's Left-Hand Rule for force direction is essential for solving related problems.

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Key Concepts

Magnetic Field Direction (Right-Hand Thumb Rule)

When a current flows through a straight wire, it generates a magnetic field. To find the direction of this…

Force Direction (Fleming's Left-Hand Rule)

Once you know the direction of the magnetic field ( $\vec{B}$ ) and the direction of the current ( $I$ ) in the…

Force per Unit Length Formula

The magnitude of the force between two long, parallel wires is typically expressed as force per unit length,…

Magnetic Field (Straight Wire): —

B = \frac{\mu_0 I}{2\pi r}

(Right-Hand Thumb Rule for direction)

Force on Current in Field: —

\vec{F} = I(\vec{L} \times \vec{B})

(Fleming's Left-Hand Rule for direction)

Force per Unit Length (Parallel Wires): —

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

Same Direction Currents: — Attract

Opposite Direction Currents: — Repel

$\mu_0$ (Permeability of Free Space): —

4\pi \times 10^{-7}\,\text{T}\cdot\text{m/A}

Ampere Definition: — Based on force between parallel currents (

2 \times 10^{-7}\,\text{N/m}

1,\text{m}

separation for

1,\text{A}

each).

Same Direction Attracts, Opposite Direction Repels (SDAR, ODAR). For the formula, remember 'Mu-naught I-one I-two over two-pi-d' sounds like a rhythmic chant for $\frac{\mu_0 I_1 I_2}{2\pi d}$ .