10. Ampere's Law

The professor begins by revisiting Ampere's law, which relates the integrated magnetic field around a closed loop to the currents passing through the loop. The law is crucial for understanding magnetostatics. The convention for the line integral is counterclockwise, with a positive current being outside the loop and a negative current inside. The law accounts for the algebraic sum of currents, not the absolute value. The subtlety of the current penetrating a surface bounded by the loop is also discussed, emphasizing that the surface's shape does not affect the outcome, as long as it has the same boundary loop.

The professor illustrates how symmetry can simplify problems involving Ampere's law, such as finding the magnetic field around a long straight wire. By considering the field's symmetry, one can deduce the field's configuration and use Ampere's law to calculate the magnetic field's magnitude. The example of a thick wire with finite thickness is also discussed, showing how the magnetic field varies within and outside the wire.

The focus shifts to the solenoid, a coil of wire that generates a magnetic field. The professor explains how to use Ampere's law to find the magnetic field inside and outside the solenoid. By considering an Amperian loop within the solenoid, it is argued that the magnetic field is constant inside and zero outside for an ideal solenoid. The right-hand rule is used to determine the direction of the magnetic field.

The professor clarifies a common mistake regarding the current through a solenoid. The current is not simply the current in the wire; instead, it is the product of the number of turns per unit length (n) and the length of the loop (l). This results in the formula for the magnetic field inside a solenoid being μ₀ times n times the current (I). The concept of an ideal solenoid, where the magnetic field lines are perpendicular and do not intersect, is also discussed.

The toroid, a solenoid wound in the shape of a doughnut, is introduced. The professor explains that the magnetic field inside the toroid is weaker towards the outer rim and stronger towards the inner rim, with the field being zero outside the toroid. The magnetic field's formula inside the toroid is derived, showing its dependence on the radius and the number of turns in the solenoid.

The professor summarizes electrostatics and magnetostatics with Maxwell's equations, which describe the relationship between electric and magnetic fields and their sources (charges and currents). The force experienced by a charged particle in an electromagnetic field is also discussed, emphasizing the importance of understanding these fields as they are responsible for the motion of charged particles.

The scenario of a wire loop moving through a magnetic field is explored, leading to the light bulb glowing due to the induced electromotive force (emf). The professor explains that the work done by the magnetic field on the moving charges within the loop results in an emf, which can be calculated using the force exerted by the magnetic field on the moving charges.

The professor discusses the apparent paradox of magnetic fields doing work despite the established principle that they do not. The resolution lies in understanding that while the magnetic field does not do work in the sense of moving charges along the direction of the magnetic field, it does facilitate the transfer of energy from macroscopic mechanical power to microscopic energy in the electrons. This is illustrated through the example of a generator, where the work done by pulling the loop results in the light bulb glowing.

The professor presents a thought experiment involving a light bulb glowing due to a magnetic field. The paradox arises when considering the situation from the frame of reference of the loop, where the loop is stationary, and the magnetic field is moving. The solution to this paradox involves the concept of a changing magnetic field inducing an electric field, which can drive a current around the loop, even in the stationary frame of reference.

The professor explains that a changing magnetic field induces an electric field, which can be understood through Faraday's law of electromagnetic induction. This law states that the electromotive force (emf) around a closed loop is equal to the negative rate of change of the magnetic flux through that loop. The induced electric field is not conservative and can drive a current around the loop, even in the absence of a conducting wire.

Lenz's law is introduced, which describes the direction of the induced current in response to a changing magnetic flux. The induced current will flow in a direction that opposes the change in magnetic flux. This is demonstrated through examples, showing how the induced current can either increase or decrease the flux, depending on the situation. Lenz's law is a fundamental principle that helps determine the direction of induced currents in electromagnetic induction scenarios.

The professor delves into the mathematical relationship between the rate of change of magnetic flux and the induced electromotive force (emf). By considering the flux through a surface that changes over time due to the motion of a loop and the change in the magnetic field, the professor derives an expression for the emf that accounts for both the changing magnetic field and the motion of the loop. This leads to a deeper understanding of Faraday's law of electromagnetic induction.

The professor concludes with Faraday's law, which establishes an intrinsic relationship between electric and magnetic fields. The law states that the line integral of the electric field around a closed loop is equal to the negative rate of change of the magnetic flux through that loop. This law reveals that a changing magnetic field can produce a circulating electric field, even in the absence of a conducting loop, and is a fundamental principle of electromagnetism.