Force due to Secondary Magnetic Field

The concept of a secondary magnetic field arises in the context of electromagnetic induction, where a changing magnetic field induces an electromotive force (EMF) and, consequently, a current in a nearby conductor. This induced current itself generates a magnetic field, which is referred to as the secondary magnetic field. The interaction between the original (primary) magnetic field and the secondary magnetic field can result in forces acting on the conductors.

Understanding Secondary Magnetic Fields

When a conductor is placed in a changing magnetic field, according to Faraday's law of electromagnetic induction, an EMF is induced in the conductor. This induced EMF can cause a current to flow if the conductor is part of a closed circuit. The current generated in the conductor due to the induced EMF then produces its own magnetic field around the conductor, which is the secondary magnetic field.

The secondary magnetic field will interact with the primary magnetic field, and according to Lenz's law, the direction of the induced current will be such that it opposes the change in the magnetic flux that produced it. This opposition can result in a force exerted on the conductor, which can be explained by the Lorentz force law.

Lorentz Force Law

The Lorentz force law describes the force experienced by a charged particle moving in a magnetic field. The force on a small segment of the conductor carrying current (I) with length (d\vec{l}) in a magnetic field (\vec{B}) is given by:

[ d\vec{F} = I \, d\vec{l} \times \vec{B} ]

where (d\vec{F}) is the differential force on the segment of the conductor, (I) is the current, (d\vec{l}) is the vector representing the small segment of the conductor, and (\vec{B}) is the magnetic field vector.

Table of Differences and Important Points

Aspect	Primary Magnetic Field	Secondary Magnetic Field
Source	External magnet or current	Induced current in the conductor
Cause	Direct application of current or magnet	Change in magnetic flux due to the primary field
Direction	Depends on the source	Opposes the change in flux according to Lenz's law
Force Interaction	Can exert a force on charges and currents	Can exert a force on the source of the primary field
Calculation	Based on the source's geometry and current	Calculated using Faraday's and Ampere's laws

Examples

Example 1: Induced Current in a Loop

Consider a circular loop of wire placed in a magnetic field that is increasing in strength. The increasing magnetic field induces a current in the loop. This induced current generates a secondary magnetic field that opposes the increase in the primary magnetic field. The force due to the secondary magnetic field can cause the loop to experience a force that might, for instance, push it out of the region of increasing magnetic field strength.

Example 2: Railgun

A railgun consists of two parallel conductive rails connected by a moving conductive armature. When a current is passed through this setup, the armature experiences a force due to the interaction between the magnetic fields (primary field due to the rails and secondary field due to the current in the armature) and accelerates along the rails. The force can be calculated using the Lorentz force law.

Conclusion

The force due to a secondary magnetic field is a fundamental concept in electromagnetism that has practical applications in various technologies, including electric motors, generators, and magnetic levitation systems. Understanding the interaction between primary and secondary magnetic fields is crucial for designing and analyzing systems that rely on electromagnetic induction.