Chapter 8: Problem 42

Find the mutual inductance between two filaments forming circular rings of radii $a$ and $\Delta a$, where $\Delta a \ll a$. The field should be determined by approximate methods. The rings are coplanar and concentric.

Short Answer

Expert verified

Question: Determine the mutual inductance between two coplanar and concentric circular rings with radii $a$ and $a+\Delta a$, where $\Delta a \ll a$. Answer: The mutual inductance between the two filaments forming circular rings of radii $a$ and $\Delta a$ is approximately $\mu_0 \Delta a$.

Step by step solution

Determine the magnetic field produced by each ring

To determine the magnetic field produced by each ring, we'll start with Ampere's law: $$B \times 2\pi r = \mu_0 I$$ where $B$ is the magnetic field, $r$ is the distance from the center of the ring, $\mu_0$ is the permeability of free space, and $I$ is the current flowing through the ring. We can solve for $B$: $$B = \frac{\mu_0 I}{2\pi r}$$ Now, we have the magnetic field as a function of $r$ for each ring.

Calculate the flux through the second ring due to the first ring

The magnetic flux through the second ring due to the magnetic field produced by the first ring is given by: $$\phi_{21} = \int_{a}^{a+\Delta a} B_1(r) 2\pi r dr$$ where $B_1(r)$ is the magnetic field produced by the first ring at a distance $r$. Using the expression for $B$ from Step 1 and assuming the same current $I$ flows through both rings, we get: $$\phi_{21} = \int_{a}^{a+\Delta a} \frac{\mu_0 I}{2\pi r} 2\pi r dr$$ $$\phi_{21} = \mu_0 I \int_{a}^{a+\Delta a} dr$$

Evaluate the integral and simplify

Evaluate the integral in the expression for $\phi_{21}$: $$\phi_{21} = \mu_0 I \left[ r \right]_a^{a+\Delta a}$$ $$\phi_{21} = \mu_0 I \left( (a+\Delta a) - a \right)$$ Since $\Delta a \ll a$, we can simplify as: $$\phi_{21} \approx \mu_0 I \Delta a$$