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.
31.3:
Calculation of Self-inductance
The self-inductance of a circuit, often simply called the inductance, is a purely geometric factor that depends only on the circuit component's structure. More specifically, it depends on the shape and size of the component that lets the flux pass through it, thus inducing an electric field that opposes any current passing through it.
Since the effect of the induced electric field and the back EMF generated depends on the rate of change of current and the self-inductance, the inductance calculation for specific geometries is required.
The strategy is to start from the magnetic field, integrate it over the area through which it passes to derive the flux, and then add up all the independent fluxes to derive the total flux. Since inductance is defined as the ratio of the total flux and the current, its formula is derived by evaluating the ratio.
For an ideal solenoid, a cylindrical toroid, and a rectangular toroid, the magnetic field is derived by assuming no edge effects; the field is assumed to be uniform inside it. In practice, this assumption is not valid. However, if the total length of the solenoid or toroid is much larger than its radius and cross-sectional area, the correction introduced is small and negligible.
It is also assumed that no magnetic field leakage is outside the system's geometry. This is also not strictly valid but approximately correct if the total length is much larger than the respective cross-sectional area.
The self-inductance of a solenoid, a cylindrical toroid, and a rectangular toroid are evaluated under these assumptions. It is noted that since each loop has the same flux passing through it and the different loops are independent, each loop contributes the same amount to the total flux. This observation throws up a factor of N2 in the formula for self-inductance. The other factors depend on the dimensions.
As a result, it is noted that whatever the dimensions, the self-inductance of a looped coil of wire can be increased by simply increasing the number of turns.
A corollary of this observation is that the self-inductance of a standard current-carrying loop of wire is much smaller than that of the coiled systems. Hence, the inductance of a standard current-carrying wire is negligible compared to inductors used as separate circuit components.
Suggested Reading
- OpenStax. (2019). University Physics Vol. 2. [Web version]. Retrieved from 14.2 Self-Inductance and Inductors – University Physics Volume 2 | OpenStax; p 635