Magnetism as Relativity: The Moving Wire

The Intuitive Picture

Imagine a long, straight wire carrying a steady electric current. In the laboratory frame, the wire is neutral: it contains positive ions fixed in the lattice and negative electrons drifting slowly. A test charge moving parallel to the wire experiences a magnetic force. Now, imagine jumping onto a "surfboard" moving at the same speed as the electrons. In your frame, the electrons are stationary. Since magnetic forces vanish for stationary charges, you would expect no force. However, due to length contraction, the spacing between the positive ions (which are moving relative to you) changes differently than the spacing between the electrons (which are stationary relative to you). This creates a net charge density in the wire. The test charge now feels a purely electric force. The magnetic force in the lab is thus revealed to be the electric force in the electron's rest frame, distorted by relativistic kinematics.

Relativistic Charge Density

To quantify this, consider a wire with positive ion density \( \lambda_+ \) and electron density \( \lambda_- \) in the lab frame. The wire is neutral, so \( \lambda_+ + \lambda_- = 0 \). Let the electrons drift with velocity \( v \) in the \( -x \) direction. We transform to a frame \( S' \) moving with velocity \( v \) in the \( -x \) direction relative to the lab. In \( S' \), the electrons are at rest. The positive ions move with velocity \( v \) in the \( +x' \) direction.

Charge density is the time component of the four-current \( J^\mu = (c\rho, \vec{J}) \). Under a Lorentz boost the charge density transforms as \( \rho' = \gamma(\rho - v J_x/c^2) \), but the cleanest route is through proper densities. The ions are at rest in the lab, so the lab value \( \lambda_+ \) is their proper density; in \( S' \) they move, and length contraction raises it to \( \lambda_+' = \gamma\lambda_+ \). The electrons move in the lab, so the lab value is already contracted: \( \lambda_- = \gamma\lambda_{-,0} \), where \( \lambda_{-,0} \) is their proper density. In \( S' \) they are at rest, so \( \lambda_-' = \lambda_{-,0} = \lambda_-/\gamma \). With \( \lambda_- = -\lambda_+ \), the net line charge in \( S' \) is \[ \lambda' = \gamma\lambda_+ - \frac{\lambda_+}{\gamma} = \lambda_+\left(\gamma - \frac{1}{\gamma}\right) = \gamma\beta^2\lambda_+ \gt 0, \] small, but not zero. This net charge creates a radial electric field in \( S' \).

Field Transformation Consistency

We can verify this result using the field transformation laws of the previous lecture. In the lab frame \( S \), the magnetic field at a distance \( r \) from the wire is \( B = \frac{\mu_0 I}{2\pi r} \) and the electric field is \( E = 0 \). The test charge moves with velocity \( v \) parallel to the wire. The magnetic force is \( F = q v B \). In the frame \( S' \) moving with the charge, the charge is at rest, so the magnetic force is zero. The force must be purely electric: \( F' = q E' \). According to the transformation law for the perpendicular electric field (assuming the boost is along the wire, and the field is radial, so perpendicular to the boost), \( E' = \gamma (E + \vec{v} \times \vec{B}) \). Since \( E=0 \) in the lab, \( E' = \gamma (\vec{v} \times \vec{B}) \). The magnitude is \( E' = \gamma v B \). The force in \( S' \) is \( F' = q \gamma v B \). Relativistic dynamics tells us that the transverse force transforms as \( F' = \gamma F \) (a transverse force on a particle at rest in \( S' \) is larger by \( \gamma \) than the force measured in a frame where the particle moves). Thus, \( F' = \gamma (q v B) \). This matches the electric force calculation \( q E' = q \gamma v B \). The consistency confirms that the magnetic force is the relativistic manifestation of the electric force in the moving frame.

The Role of Length Contraction

The physical origin of the net charge density in \( S' \) is length contraction. In the lab, the wire is neutral. The proper length of the electron segment is contracted by \( \gamma \) relative to its rest frame. In \( S' \), the electrons are at rest, so their spacing is the proper spacing. The ions, however, are moving in \( S' \), so their spacing is contracted by \( \gamma \) relative to the lab spacing. Since the lab spacing was equal for both, the ion spacing in \( S' \) is smaller than the electron spacing. This results in a higher density of positive charge than negative charge in \( S' \). The electric field generated by this net charge density is exactly the transformed field predicted by the tensor formalism. This demonstrates that magnetism is a kinematic effect arising from the relativity of simultaneity and length contraction.