Both circular coils A and B (Fig. 𝐄 27 . 4 0 ) have area A and N turns. They are free to rotate

Both circular coils A and B (Fig. 𝐄 27 . 4 0 ) have area A...

Both circular coils A and B (Fig. $\mathbf{E} 27 . \mathbf{4 0}$ ) 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 $\overrightarrow{\boldsymbol{B}}$ in the $+z$ -direction. (a) What is the direction of the magnetic moment $\overrightarrow{\boldsymbol{\mu}}$ 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

Equilibrium Stability

Stability in the context of magnetic dipoles refers to whether small deviations from the equilibrium orientation result in forces or torques that restore the original state. An equilibrium is considered stable if small perturbations lead to an increase in potential energy, which then acts to return the system to the equilibrium configuration. Conversely, if a small displacement reduces the potential energy, the equilibrium is unstable as there is no restorative force to bring the system back.

Magnetic Potential Energy

The potential energy associated with a magnetic dipole in a magnetic field is defined by the expression U = -? · B. This energy quantifies the work done in bringing the dipole from a reference orientation (typically perpendicular to the field) to a given orientation relative to the field. Lower potential energy indicates a more stable alignment, with the minimum energy configuration occurring when the dipole is aligned with the field.

Magnetic Dipole Moment

The magnetic dipole moment is a vector quantity associated with a current loop, determined by the product of the current, the number of turns, and the area of the loop. Its direction is given by the right?hand rule, where curling the fingers in the direction of current flow makes the thumb point in the direction of the magnetic moment. This concept is fundamental for understanding the behavior of current loops in external magnetic fields.

Torque on a Magnetic Dipole

The torque experienced by a magnetic dipole in a magnetic field is given by the cross product of the magnetic moment and the magnetic field, expressed as ? = ? × B. This torque tends to rotate the dipole to align its magnetic moment with the magnetic field. However, when the dipole is oriented such that ? is parallel or antiparallel to B, the torque is zero since the cross product vanishes, leading to a state of rotational equilibrium.

Transcript

00:01 Hi, in the given problem there are two identical circular coils which are having the same area of cross -section and same number of turns in them and carrying the same magnitude of electric current.

00:23 So these are the coils, the circular coils, coil a and coil b.

00:33 This is coil a, this is coil a, this is.

00:35 Is coil b.

00:37 Both are having their diameters along x -axis and y -axis.

00:47 This first coil a is carrying the current in clockwise direction and coil b is carrying the same current but in counter -clockwise direction.

00:59 Area of each coil is given to be a and number of terms in each one of them is equal to n both of these coils can be rotated about x -axis now in the first part of the problem we have to find the directions of magnetic moments associated with these coils and we know magnetic moments the directions of magnetic moments are always from south pole to north pole now here upper phase of coil a behaves like south pole as the direction of current in it is clockwise so it will behave like south pole so the lower phase which is not visible to us will be behaving like a north pole hence the direction of magnetic field magnetic movement so the magnetic moment of this coil will be from south to north means will be into the plane of paper.

02:56 And for coil a, sorry, for coil b, as the upper face is carrying the current in counterclockwise direction.

03:08 So its upper face will behave like north.

03:20 Pole and as the direction of magnetic moment is from south to north so it will be coming up.

03:27 So magnetic moment of coil b will be coming out perpendicular to the plane of paper.

03:47 This is the answer for the first part of the problem.

03:50 Now in the second part of the problem we have to show that these two coils are in rotational equilibrium for which we will find the torque experienced by these two coils.

04:03 So torque experienced by coil a will be given by in vector form this is given as the cross product of magnetic moment with the magnetic field.

04:15 As the direction of magnetic field for both of these coils is coming out perpendicularly to the plane of paper as it is given that the magnetic field is along positive z -axis means for both the coils the magnetic field is coming out so for coil a as the magnetic moment is inward into the plane of paper and magnetic field is outward out of the plane of paper so if i use the rule for the cross product here it comes out to be mu into b into sine theta and that theta will be one eight -factor and that theta will be one so it comes out to be mu into b into sine 180 degree...

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