Both circular coils A and B (Fig. E27.44) have area A and N turns. They are free to rotate about

step 1 This is chapter 27 problem number 44. We have two coils. One is carrying current, both -wise cur

Both circular coils A and B (Fig. E27.44) have area A and N...

Both circular coils A and B ($\textbf{Fig. E27.44}$) have area A and N turns. They are free to rotate about a diameter that coincides with the $x-$axis. Current $I$ circulates in each coil in the direction shown. There is a uniform magnetic field $\overrightarrow{B}$ in the $+z$-direction. (a) What is the direction of the magnetic moment $\overrightarrow{\mu}$ for each coil? (b) Explain why the torque on both coils due to the magnetic field is zero, so the coil is in rotational equilibrium. (c) Use Eq. (27.27) to calculate the potential energy for each coil. (d) For each coil, is the equilibrium stable or unstable? Explain.

Key Concepts

Magnetic Dipole Moment

The magnetic dipole moment is a vector quantity that represents the strength and orientation of a current loop’s magnetic field. It is defined for a coil with multiple turns as the product of the number of turns, the current, and the area of the loop, with its direction determined by the right-hand rule. This concept is fundamental in understanding how the coil interacts with an external magnetic field.

Torque on a Magnetic Dipole

A current-carrying loop in an external magnetic field experiences a torque given by the cross product of its magnetic moment and the magnetic field (? = ? × B). This torque tends to rotate the dipole so that its magnetic moment aligns with the magnetic field, and understanding this interaction is key to analyzing the rotational behavior of magnetic systems.

Potential Energy of a Magnetic Dipole

The potential energy associated with a magnetic dipole in a magnetic field is given by U = -? ? B. This energy is a measure of the work done by the magnetic field on the dipole and is minimized when the dipole is aligned with the field. This concept is crucial for determining the energy states and equilibrium positions of the dipole in a magnetic field.

Rotational Equilibrium and Stability

Rotational equilibrium occurs when the net torque acting on an object is zero, implying that it does not spontaneously start rotating. The stability of this equilibrium depends on whether small perturbations increase or decrease the potential energy; if a small displacement results in an increase in potential energy, the equilibrium is stable, whereas if it results in a decrease, the equilibrium is unstable. This concept is important for assessing whether a system will return to or deviate from its equilibrium state upon slight disturbances.

Transcript

00:01 This is chapter 27 problem number 44.

00:03 We have two coils.

00:06 One is carrying current, both -wise current.

00:09 We call this coil a.

00:12 When we have a melancho that has current along the opposite direction, coil b.

00:18 The magnetic field is out of the page.

00:23 So the green is the magnetic field b.

00:28 In part a, we are actually asked to determine the direction of the magnetic dipole moment vector.

00:37 It's going to be equal to n -i times a as you know.

00:42 So from right -hand rule by aligning your fingers along the direction of the current, your thumb is going to show the direction of a magnetic moment vector.

00:56 So for a, then and mu is into the page where b is going to be the direction of the magnetic back hole moment is going to be out of the page, again, from the right handle.

01:17 In court b, we are asked to calculate the torque, net torque, on each coil.

01:24 So again, the magnitude of the torque in terms of the mechanics.

01:29 Moment and the magnetic field is the vector product, cost product of mu and b vectors.

01:39 If you're going to write the magnitude of the torp, then it's mu b, sine, theta.

01:46 Then we're going to have to determine the angle between mu and b for each case.

01:53 So for a, for example, we said that the direction of mu is in the direction of mu is in.

02:01 Into the page, but we know that the direction of b, sorry, i'm not to make this green, this is the direction of b no matter what...

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