A toroid having a rectangular cross-section (a= 2.00 cm by...

A toroid having a rectangular cross-section $(a=$ $2.00 \mathrm{~cm}$ by $b=3.00 \mathrm{~cm})$ and inner radius $R=4.00 \mathrm{~cm}$ consists of 500 turns of wire that carries a current $I=I_{\max } \sin \omega t$, with $I_{\max }=50.0 \mathrm{~A}$ and a frequency $f=\omega / 2 \pi=60.0 \mathrm{~Hz}$. A coil that consists of 20 turns of wire links with the toroid, as shown in Figure P31.17. Determine the emf induced in the coil as a function of time.

A toroid having a rectangular cross-section $(a=$ $2.00 \mathrm{~cm}$ by $b=3.00 \mathrm{~cm})$ and inner radius $R=4.00 \mathrm{~cm}$
consists of 500 turns of wire that carries a current $I=I_{\max } \sin \omega t$, with $I_{\max }=50.0 \mathrm{~A}$ and a frequency
$f=\omega / 2 \pi=60.0 \mathrm{~Hz}$. A coil that consists of 20 turns of wire links with the toroid, as shown in Figure P31.17. Determine the emf induced in the coil as a function of time.

Key Concepts

Faraday's Law of Electromagnetic Induction

This law states that a changing magnetic flux through a loop or coil induces an electromotive force (emf) in it. It is fundamental to understanding how time-varying magnetic fields create currents, and it is mathematically expressed as the negative rate of change of magnetic flux through the coil. This concept is key to solving problems involving transformers, inductors, and any system where a magnetic field interacts with conductive loops.

Magnetic Field in a Toroid

A toroid is a doughnut-shaped coil where the magnetic field is largely confined within its core, and its magnitude can be derived using Ampere's law. The field inside the toroid varies inversely with the distance from the center and is nearly zero outside the core. Understanding this field distribution is essential for calculating the magnetic flux through any coil linked with the toroid.

Magnetic Flux Calculation

Magnetic flux is defined as the product of the magnetic field and the area through which it penetrates, taking into account the angle between the field and the surface normal. In problems involving a toroid with a rectangular cross-section, the flux is found by integrating the magnetic field over the cross-sectional area. Accurately determining the flux is critical, as its time derivative leads directly to the value of the induced emf.

Time-Varying (Sinusoidal) Current

When the current in a circuit varies sinusoidally with time, it causes the magnetic field and consequently the magnetic flux to also vary sinusoidally. This time dependence is significant because the induced emf in any linked coil depends on the rate at which the magnetic flux changes. The sinusoidal nature of the driving current leads to an induced emf that also varies sinusoidally in time, typically requiring the use of differentiation of trigonometric functions.

Mutual Inductance between Coupled Coils

Mutual inductance quantifies the ability of one circuit (or coil) to induce an emf in another circuit due to a change in current in the first circuit. In scenarios where a coil is linked with another source (like a toroid), the mutual inductance characterizes the efficiency with which the changing magnetic flux is transferred, thereby helping to predict the magnitude and phase of the induced emf in the secondary coil.

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