31.3
Calculation of Self-inductance
Inside an ideal cylindrical solenoid with N number of turns, length l, and cross-sectional area A, the uniform magnetic field is known. The magnetic flux through any turn is derived, and the total flux is calculated. The definition of self-inductance then gives its formula.
When the solenoid is wrapped into a circle, it becomes a toroid of radius r. Then, l is the circumference, which gives its inductance.
If its cross-section is a rectangle, it is called a rectangular toroid of height h.
The magnetic field is the same. The flux through any loop is the differential flux integrated from its inner radius to its outer radius. Noting the height is constant, the integral is evaluated. So, the total flux is derived, which gives its self-inductance.
Recall the self-inductance of a current-carrying wire of radius R. Compare it to the solenoid's self-inductance.
The ratio depends on geometric terms and the factor N2, also for toroids.
This factor makes the self-inductance of any coiled system much larger than a current-carrying wire's self-inductance, which is neglected.
The self-inductance of a circuit, often simply called the inductance, is a purely geometric factor that depends only on the circuit component's struct…
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