Question

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

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: $$\phi_{21} \approx \mu_0 I \Delta a$$

Find the mutual inductance

The mutual inductance \(M\) between the two rings is given by the ratio of the flux through the second ring to the current in the first ring: $$M = \frac{\phi_{21}}{I}$$ Using our expression for \(\phi_{21}\) from Step 3, we find: $$M \approx \mu_0 \Delta a$$ The mutual inductance between the two filaments forming circular rings of radii \(a\) and \(\Delta a\) is approximately \(\mu_0 \Delta a\).

Key concepts

Ampere's law

Ampere's Law is a fundamental concept in electromagnetism that relates the magnetic field around a closed loop to the electric current passing through the loop. It is mathematically expressed as: \[ B \times 2\pi r = \mu_0 I \] where:

• \( B \) is the magnetic field.
• \( r \) is the distance from the center.
• \( \mu_0 \) is the permeability of free space, a constant.
• \( I \) is the current passing through the loop.

Using Ampere's Law, you can determine the magnetic field (\( B \)) around a conductor by rearranging the formula: \[ B = \frac{\mu_0 I}{2\pi r} \] This is particularly useful for cases involving symmetrical current arrangements, like circular rings. For each of the rings in the exercise, Ampere's Law allows us to find how the current flowing in one impacts the space surrounding it.

Magnetic field

The magnetic field is a vector field around a magnetic material or a moving electric charge within which magnetic force is exerted. From Ampere's Law, we learned that the magnetic field due to a current-carrying ring at a distance \( r \) is calculated by substituting values into the equation: \[ B = \frac{\mu_0 I}{2\pi r} \] For circular rings, this equation provides the magnetic field strength as it varies with the position \( r \) around the ring. Magnetic fields are crucial for understanding how different objects exert forces on one another due to magnetism. In scenarios involving circular rings, the field is strongest at the points closest to the current and decreases as you move farther away. When evaluating mutual inductance, understanding the formation and distribution of the magnetic field around the ring is key to finding the flux through adjacent loops.

Flux

Magnetic flux represents the measure of the quantity of magnetism, taking into account the strength and the extent of the magnetic field. It is relevant in determining mutual inductance—which assesses how a change in current in one conductor causes a voltage across a second conductor. The flux (\( \phi_{21} \)) through one ring due to another is given by integrating the magnetic field over the area:\[ \phi_{21} = \int_{a}^{a+\Delta a} B_1(r) 2\pi r \, dr \] Substituting for the magnetic field \( B \) using \( \frac{\mu_0 I}{2\pi r} \), we find:\[ \phi_{21} = \mu_0 I \int_{a}^{a+\Delta a} dr \] Evaluating this integral provides a practical way of calculating the induced interactions between the rings. In the given problem, it simplifies to \( \phi_{21} \approx \mu_0 I \Delta a \), which helps further to solve for mutual inductance.

Circular rings

Circular rings are commonly used geometric structures in electromagnetic studies due to their symmetry and simplicity in calculations. These rings, when carrying current, produce magnetic fields around them, akin to those of a magnetic dipole. In our exercise, two rings are considered:

• Ring with radius \( a \).
• Ring with radius smaller by \( \Delta a \), where \( \Delta a \ll a \).

Being coplanar and concentric, these rings align their magnetic fields in such a way that we can study the mutual influence termed as mutual inductance. This is an important concept showing how changes in electrical properties of one affect the other. For the two close rings, we found out that the mutual inductance is influenced directly by the size of the smaller ring multiplied by the magnetic permeability: \[ M \approx \mu_0 \Delta a \] Understanding the basic dynamics of circular rings helps simplify complex electromagnetic problems into understandable solutions.