Mutual Inductance of Two Concentric Loops at an Angle: 04-BS-9, December 2014, Q4

Question 4: Mutual Inductance of Two Concentric Loops at an Angle (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given.

Quantity	Value
Small loop radius $a$	$1\times10^{-3}$ m
Large loop radius $b$	$3\times10^{-2}$ m
Angle between loop planes $\theta$	$30^\circ$

Find. The mutual inductance $M$ between the two loops.

Small loop (radius a) at the centre of a large loop (radius b), its plane tilted 30° from the large loop's plane.

Approach. Since $a\ll b$, treat the field the large loop produces at its own centre as uniform over the small loop's area. The flux linking the small loop is that field times the small loop's area, projected by $\cos\theta$ onto the small loop's normal (the angle between the two loop planes equals the angle between their normals).

Field at the centre of the large loop (per unit current). $$B_{\text{centre}}=\frac{\mu_0 I}{2b}$$
Flux through the small loop and mutual inductance. With $\theta=30^\circ$ between the loop planes (equal to the angle between their normals): $$M=\frac{\Phi_{\text{small}}}{I}=\frac{\mu_0\,\pi a^2\cos\theta}{2b}=\frac{(4\pi\times10^{-7})\,\pi(1\times10^{-3})^2\cos30^\circ}{2(0.03)}$$ $$M=\boxed{5.70\times10^{-11}\ \text{H}=57.0\ \text{pH}}$$

Quantity	Result
$M$	$5.70\times10^{-11}$ H (57.0 pH)

Concept & Theory

Topic. Mutual inductance of two concentric loops of very different size, using the "small loop in a locally-uniform field" approximation.

Key relations. Field at the centre of a circular loop, $B=\mu_0 I/2b$; mutual inductance $M=\Phi_{\text{small}}/I$; flux through a tilted loop, $\Phi=BA\cos\theta$.

Why this works. Because $a\ll b$, the field the large loop produces varies negligibly over the small loop's tiny area, so it can be treated as the uniform value at the common centre — this is exactly the "suitable approximation" the question asks for, and avoids a full elliptic-integral mutual-inductance calculation. Reciprocity ($M_{12}=M_{21}$) means it is far easier to compute the small loop's flux due to the large loop's field than the reverse.

Common pitfall. Using the angle between the field direction and the small loop's plane ($\sin\theta$) instead of between the field and the loop's normal ($\cos\theta$).

Source. Sadiku, Elements of Electromagnetics, Ch. 9 (Mutual inductance of coupled circuits).

Question 4 of 8: Mutual Inductance of Two Concentric Loops at an Angle

Question 4: Mutual Inductance of Two Concentric Loops at an Angle (20 marks)