An electric dipole with dipole moment p⃗ is in a uniform external electric field E. (a) Find the

An electric dipole with dipole moment p⃗ is in a uniform...

An electric dipole with dipole moment $\vec{p}$ is in a uniform external electric field $E$. (a) Find the orientations of the dipole for which the torque on the dipole is zero. (b) Which of the orientations in part (a) is stable, and which is unstable? (Hint: Consider a small rotation away from the equilibrium position and see what happens.) (c) Show that for the stable orientation in part (b), the dipole's own electric field tends to oppose the external field.

Key Concepts

Electric Dipole

An electric dipole consists of two equal and opposite charges separated by a small distance, and its behavior in an electric field is characterized by its dipole moment, a vector pointing from the negative charge to the positive charge. Understanding the dipole moment is crucial for analyzing how the dipole interacts with external electric fields and for determining the torque and potential energy associated with its orientation.

Torque on a Dipole in an Electric Field

When placed in an external electric field, a dipole experiences a torque that tends to align it with the field. This torque is given by the cross product of the dipole moment and the electric field, and its magnitude is proportional to the sine of the angle between these vectors. Analyzing this torque is key to determining the orientations at which the net torque vanishes.

Equilibrium Orientations

Equilibrium occurs when the net torque on the dipole is zero, which happens when the dipole is aligned either parallel or anti-parallel to the external electric field. Identifying these orientations is essential for understanding the conditions under which the dipole remains static in the field before considering perturbations.

Stability Analysis of Equilibrium

Assessing the stability of an equilibrium orientation involves examining how the system behaves under small perturbations from that orientation. By considering small rotations and using the potential energy associated with the dipole (typically given by the negative dot product of the dipole moment and the electric field), one can determine which equilibrium configuration corresponds to a minimum (stable) or a maximum (unstable) in the potential energy landscape.

Opposition of the Dipole's Field to the External Field

In the stable equilibrium configuration, specifically when the dipole is aligned parallel to the external field, the dipole's own electric field counteracts the external field. This opposition arises from the nature of the induced field generated by the dipole and plays a significant role in the overall energy and dynamics of the system by effectively reducing the net field experienced in the vicinity of the dipole.

Transcript

00:01 So torque acting on a dipole in presence of an electric field is given by magnitude of the dipole times magnitude of the electric field times sine of the angle between them.

00:19 Now if the net torque is zero, this means that theta has to be either zero degree or 180 degree.

00:33 And this is for the case when you have a dipole and hence the electric field and the magnetic dipole, sorry, the electric field and the electric dipole moment are not zeros.

00:46 So for that case, only possibility that for the torque to be zero is when the value of theta is either zero degree or 180 degree, which means that the dipole moment should either be parallel, or anti -parallel to the electric field.

01:07 So let's draw the orientations of the dipole in a uniform electric field.

01:14 So let's say the electric field is along towards the right or towards the positive x -axis if this is your x -axis.

01:28 Now the condition that we got is the dipole should has to be either parallel or parallel to the electric field in order to in order for the net torque to be zero.

01:42 So if the dipole is parallel to the electric field, then we should have the negative charge to be on the left and positive charge on the right, since dipole always points towards the positive charge.

02:02 Now the other possibility is for the dipole to be anti -parallel to the electric fields.

02:07 For that the direction of the dipole or the dipole moment should be like so and for this case we should have the negative charge on the right and positive charge on the left so these are the two orientations for the dipole so that the torque on the dipole is zero now in the second part we have to find which orientation of the dipole is stable now the this is really simple.

02:41 The stable orientation is when the dipole is parallel to the electric field.

02:48 So theta should be equal to zero for stability.

02:53 And unstable orientation is the one where both of them are anti -parallel...

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