A copper ring of radius a is at a fixed distance 𝐝 (with a<<d ) directly above an identical copp

step 1 Okay, as per the question, two concentric coils, each of the radius equals to 2 pi centimeter. O

A copper ring of radius a is at a fixed distance 𝐝 (with...

A copper ring of radius $a$ is at a fixed distance $\mathbf{d}$ (with $a<<d$ ) directly above an identical copper ring. Each ring has a resistance $R$ for circulating currents. An increasing current $I=I_0 t / \tau$ is applied in the lower ring. Neglect the self-inductance of each ring, and make appropriate approximations. (a) Find the dipole moment of the lower ring. (b) Find the magnetic flux through the upper ring. (c) Find the induced EMF and the current in the upper ring. (d) Find the induced dipole moment of the upper ring. (e) Show that the force between the rings is $$ F=\frac{12 \pi^4 a^8 I_0^2 t}{c^4 3 R d^7 \tau^2} . $$ Is this force repulsive or attractive?

A copper ring of radius $a$ is at a fixed distance $\mathbf{d}$ (with $a<<d$ ) directly above an identical copper ring. Each ring has a resistance $R$ for circulating currents. An increasing current $I=I_0 t / \tau$ is applied in the lower ring. Neglect the self-inductance of each ring, and make appropriate approximations.
(a) Find the dipole moment of the lower ring.
(b) Find the magnetic flux through the upper ring.
(c) Find the induced EMF and the current in the upper ring.
(d) Find the induced dipole moment of the upper ring.
(e) Show that the force between the rings is
$$
F=\frac{12 \pi^4 a^8 I_0^2 t}{c^4 3 R d^7 \tau^2} .
$$
Is this force repulsive or attractive?

Key Concepts

Magnetic Dipole Moment

The magnetic dipole moment represents the strength and orientation of a magnetic source generated by a circulating current. In the context of current loops, it is calculated as the product of the current and the area enclosed by the loop, and it characterizes how the loop behaves in an external magnetic field, such as by producing or responding to torque and force.

Magnetic Flux

Magnetic flux quantifies the total magnetic field passing through a specified area. It is determined by integrating the component of the magnetic field perpendicular to the surface over the area, and it is central to understanding how changes in the magnetic field can induce an electromotive force in nearby conductors.

Faraday's Law of Induction

Faraday's law states that a changing magnetic flux through a circuit induces an electromotive force (EMF) in that circuit. This principle is fundamental in electromagnetism, linking time-varying magnetic fields to the production of electric currents, and forms the basis for many electrical devices and energy conversion processes.

Lenz's Law

Lenz's law is a consequence of energy conservation, stating that the induced current in a conductor will flow in such a direction that its magnetic field opposes the change in the magnetic flux that produced it. This principle explains the direction of induced currents and provides insight into the stability and dynamics of electromagnetic systems.

Mutual Inductance

Mutual inductance describes the phenomenon where a change in current in one circuit induces an EMF in a nearby circuit. It is a measure of how effectively magnetic flux produced by one conductor links with another, and plays a critical role in devices like transformers and coupled circuits.

Ohm's Law in the Context of Induced Currents

Ohm's law relates the voltage across a resistor to the current flowing through it by way of the resistance. When applied to circuits experiencing induced EMF due to changing magnetic flux, Ohm's law allows the calculation of induced currents, providing a direct link between electromagnetic induction and circuit response.

Electromagnetic Force Between Current Loops

The force between current-carrying loops can be understood in terms of the interaction of their magnetic fields. When loops with induced or permanent magnetic moments are placed in each other’s fields, gradients in the fields result in forces that can be either attractive or repulsive, depending on the orientation and nature of the currents involved.

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