(III) A toroid has a rectangular cross section as shown in Fig. 30–26. Show that the self-inductance is


$L=\frac{{\mu }_0{N}^{2}h}{2\pi }\ln \frac{r_2}{r_1}$


where N is the total number of turns and r₁, r₂ and h are the dimensions shown in Fig. 30–26. [Hint: Use Ampère’s law to get B as a function of r inside the toroid, and integrate.]

Verified step by step guidance

1. Step 1: Begin by understanding the geometry of the toroid. A toroid is a circular coil with a rectangular cross-section. The dimensions provided are r₁ (inner radius), r₂ (outer radius), and h (height of the rectangular cross-section). The total number of turns is N.
2. Step 2: Use Ampère’s law to find the magnetic field B inside the toroid. Ampère’s law states: ∮B·dl = μ₀I_enclosed, where I_enclosed is the current enclosed by the path. For a toroid, the path of integration is a circular loop of radius r within the toroid. The enclosed current is N times the current I in each turn, so I_enclosed = N·I. The magnetic field B is uniform along the circular path, and the path length is 2πr. Thus, B·(2πr) = μ₀(N·I), leading to B = (μ₀N·I) / (2πr).
3. Step 3: The self-inductance L is defined as the ratio of the magnetic flux Φ to the current I: L = Φ / I. To calculate Φ, consider the magnetic flux through the cross-sectional area of the toroid. The flux through a small strip of width dr at radius r is dΦ = B·(h·dr), where h is the height of the rectangular cross-section. Substitute B = (μ₀N·I) / (2πr) into this expression: dΦ = [(μ₀N·I) / (2πr)]·(h·dr).
4. Step 4: Integrate dΦ over the range of r from r₁ to r₂ to find the total magnetic flux Φ. The integral is Φ = ∫[r₁ to r₂] [(μ₀N·I) / (2πr)]·(h·dr). Factor out constants μ₀, N, I, h, and 2π from the integral: Φ = (μ₀N·I·h) / (2π) ∫[r₁ to r₂] (1/r) dr. The integral of 1/r is ln(r), so Φ = (μ₀N·I·h) / (2π)·[ln(r₂) - ln(r₁)].
5. Step 5: Divide Φ by I to find the self-inductance L. Since L = Φ / I, substitute Φ: L = [(μ₀N·h) / (2π)]·[ln(r₂) - ln(r₁)]. Combine the logarithmic terms using the property ln(a) - ln(b) = ln(a/b): L = [(μ₀N²·h) / (2π)]·ln(r₂ / r₁). This is the expression for the self-inductance of the toroid.

Key Concepts

Self-Inductance

Self-inductance is a property of a coil or circuit that quantifies its ability to induce an electromotive force (EMF) in itself due to a change in current. It is represented by the symbol L and is measured in henries (H). The self-inductance depends on the geometry of the coil, the number of turns, and the magnetic permeability of the core material.

Ampère’s Law

Ampère’s Law relates the integrated magnetic field around a closed loop to the electric current passing through the loop. Mathematically, it states that the line integral of the magnetic field B around a closed path is equal to μ₀ times the total current I enclosed by the path. This law is fundamental in deriving the magnetic field inside a toroid, which is essential for calculating self-inductance.

Magnetic Field in a Toroid

The magnetic field inside a toroid is generated by the current flowing through its windings and is confined within the core. The magnetic field strength B at a distance r from the center of the toroid can be derived using Ampère’s Law, leading to the expression B = (μ₀N I) / (2πr) for r between the inner and outer radii. Understanding this relationship is crucial for calculating the self-inductance of the toroid.