explain the origin of the forces between current-carrying conductors and determine the direction of the forces

1. Introduction

When an electric current flows through a conductor it produces a magnetic field in the surrounding space. This magnetic field can exert forces on other nearby currents. Understanding the origin and direction of these forces is essential for topics such as the definition of the ampere, electromagnetic devices and the operation of motors.

2. Magnetic field produced by a straight current‑carrying conductor

2.1 Biot–Savart law (qualitative form)

The magnetic field \$ \mathbf{B} \$ at a point P due to a small element \$ \mathrm{d}\mathbf{l} \$ of a current \$ I \$ is given by

\$\mathrm{d}\mathbf{B}= \frac{\mu_0}{4\pi}\frac{I\,\mathrm{d}\mathbf{l}\times\mathbf{\hat{r}}}{r^{2}}\$

where \$ \mathbf{\hat{r}} \$ is a unit vector from the element to the point and \$ r \$ is the distance between them.

2.2 Result for an infinitely long straight wire

Integrating the Biot–Savart law around an infinite straight wire gives the well‑known expression

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

where \$ r \$ is the perpendicular distance from the wire. The direction of \$ \mathbf{B} \$ follows the right‑hand grip rule: thumb along the direction of conventional current, fingers curl in the direction of the magnetic field lines.

3. Force on a current‑carrying conductor in a magnetic field

The magnetic force on a length \$ \mathbf{L} \$ of conductor carrying current \$ I \$ in a magnetic field \$ \mathbf{B} \$ is given by the Lorentz force law for a current element:

\$\mathbf{F}= I\,\mathbf{L}\times\mathbf{B}\$

Key points:

• The magnitude is \$ F = I L B \sin\theta \$, where \$ \theta \$ is the angle between \$ \mathbf{L} \$ and \$ \mathbf{B} \$.
• The direction is obtained using the right‑hand rule: point the fingers along \$ \mathbf{L} \$ (current direction), curl them toward \$ \mathbf{B} \$; the thumb points in the direction of \$ \mathbf{F} \$.

4. Interaction between two parallel current‑carrying conductors

4.1 Origin of the force

Consider two long, straight, parallel conductors separated by a distance \$ d \$. Conductor 1 produces a magnetic field \$ B_1 \$ at the location of conductor 2:

\$B1 = \frac{\mu0 I_1}{2\pi d}\$

Conductor 2, carrying current \$ I2 \$, experiences a force due to \$ B1 \$:

\$F{21}= I2 L B1 = \frac{\mu0 I1 I2 L}{2\pi d}\$

By Newton’s third law the force on conductor 1 is equal in magnitude and opposite in direction.

4.2 Determining the direction of the force

Use the right‑hand rule for each conductor:

1. Find the magnetic field produced by conductor 1 at the position of conductor 2 (circles around conductor 1).
2. Apply the \$ I2\mathbf{L}\times\mathbf{B}1 \$ rule to obtain the force on conductor 2.
3. Reverse the roles to obtain the force on conductor 1; the forces are opposite and collinear.

5. Summary of force direction for parallel conductors

6. Key take‑aways

• A current produces a circular magnetic field described by \$ B = \mu_0 I / (2\pi r) \$.
• The magnetic force on a current element is \$ \mathbf{F}=I\mathbf{L}\times\mathbf{B} \$.
• Two parallel conductors exert forces on each other because each experiences the magnetic field produced by the other.
• Same‑direction currents attract; opposite‑direction currents repel – a result of the right‑hand rule applied to each wire.
• These principles define the ampere and underlie the operation of many electromagnetic devices.