Question E6D10:
What is a primary advantage of using a toroidal core instead of a solenoidal core in an inductor?  

Choices:  
A: Toroidal cores confine most of the magnetic field within the core material  
B: Toroidal cores make it easier to couple the magnetic energy into other components  
C: Toroidal cores exhibit greater hysteresis  
D: Toroidal cores have lower Q characteristics    

Correct Answer is  A: Toroidal cores confine most of the magnetic field within the core material  

Correct answer: A — Toroidal cores confine most of the magnetic field within the core material.

Why A is correct:
- A toroid provides a closed magnetic path. By Ampere’s law, the field is largely inside the core and very small outside, which reduces stray fields, unwanted coupling, and RF radiation.
- Ideal toroid field:
  $$H(r) = \frac{N I}{2\pi r}, \quad B(r) = \mu H(r) = \frac{\mu N I}{2\pi r} \text{ (inside the toroid)}, \quad \approx 0 \text{ outside}$$
  where $N$ is turns, $I$ is current, and $\mu = \mu_0 \mu_r$.

Why the other choices are incorrect:
- B: “Easier to couple magnetic energy into other components” — False. Toroids reduce external fields, so they are harder (not easier) to couple unintentionally. Solenoids with fringing fields couple more readily.
- C: “Greater hysteresis” — False. Hysteresis is primarily a property of the core material and flux density, not the core shape. Toroids often have uniform flux and no air gap, which can reduce losses, not increase them.
- D: “Lower Q characteristics” — False. Q depends on $Q = \omega L / R_{\text{loss}}$. Toroids often achieve high Q at RF because of confined fields (low radiation and leakage). Actual Q depends on material losses, but toroids do not inherently have lower Q.

Useful formulas and rules of thumb:
- Inductance of a toroid (mean radius $r_m$, cross-sectional area $A$):
  $$L \approx \frac{\mu_0 \mu_r N^2 A}{2\pi r_m}$$
- Inductance of a solenoid (length $\ell$):
  $$L \approx \frac{\mu_0 \mu_r N^2 A}{\ell}$$
- Quality factor:
  $$Q = \frac{\omega L}{R_{\text{loss}}}$$
- Rules of thumb:
  - Use toroids to minimize EMI/RFI, mutual coupling, and radiation; good for RF chokes and compact inductors.
  - Use air-core solenoids when extremely high Q and zero core loss are needed and stray fields are acceptable.