this problem explores how a current-carrying wire can be accelerated by a magnetic field. you will use the ideas of magnetic flux and the emf due to change of flux through a loop. note that there is an involved follow-up part that will be shown once you have found the answer to part b.
How can a current-carrying wire be accelerated by a magnetic field using the concept of magnetic flux and electromotive force (emf) due to the change of flux through a loop?
Answer:
The acceleration of a current-carrying wire by a magnetic field involves the interaction of magnetic fields with the moving charge carriers in the wire. This phenomenon can be explained by Faraday’s law of electromagnetic induction, which describes how a changing magnetic field induces an electromotive force (emf) in a conductor.
- Magnetic Flux:
When a current-carrying wire is placed in a magnetic field, the magnetic field exerts a force on the moving charges within the wire, causing it to experience a Lorentz force that can accelerate the wire. The magnetic flux through a loop is the measure of the magnetic field passing through the area enclosed by the loop. Mathematically, magnetic flux Φ is given by the product of the magnetic field B, the area of the loop A, and the angle between the magnetic field and the normal to the loop:
Φ = B \cdot A \cdot \cos(θ)
- Electromotive Force (emf) due to Change of Flux:
According to Faraday’s law of electromagnetic induction, an electromotive force (emf) is induced in a conductor when there is a change in magnetic flux through the circuit. The induced emf \mathcal{E} is proportional to the rate of change of magnetic flux and is given by:
\mathcal{E} = -\frac{dΦ}{dt}
This means that when the magnetic flux through a loop changes, an emf is induced in the loop which leads to the flow of current in the wire. The direction of the induced emf follows Lenz’s Law, which states that the induced current will flow in a direction that opposes the change in magnetic flux causing it.
- Acceleration of the Wire:
When there is a change in the magnetic flux passing through the loop due to the movement of the wire in the magnetic field or a change in the magnetic field itself, an emf is induced in the wire. This induced emf leads to the flow of current in the wire, creating a magnetic force that interacts with the external magnetic field. As a result, the wire experiences a Lorentz force that accelerates the wire in a direction perpendicular to both the magnetic field and the current flow.
In conclusion, the acceleration of a current-carrying wire by a magnetic field relies on the principles of magnetic flux and electromotive force due to the change of flux through a loop. By understanding these concepts, we can explain how a current-carrying wire experiences acceleration in the presence of a magnetic field, showcasing the fascinating interplay between electricity and magnetism.