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Chapter 21: Problem 56

An electric dipole with a dipole moment of magnitude \(p\) is placed at various orientations in an electric field \(\vec{E}\) that is directed to the left. (a) What orientation of the dipole will result in maximum torque directed into the page? What then is the electric potential energy? (b) What orientation of the dipole will give zero torque and maximum electric potential energy? What type of equilibrium is this: stable, unstable, or neutral?

Short Answer

Expert verified
For maximum torque, the electric dipole should be perpendicular to the electric field direction with zero potential energy. For zero torque and maximum potential energy, the dipole should be oriented antiparallel to the field direction. This is also an unstable equilibrium state.

Step by step solution

01

Determine the orientation for maximum torque

The torque experienced by a dipole in an electric field is determined by the formula \(蟿 = pE \sin 胃\), where \(胃\) is the angle between \(\vec{p}\) and \(\vec{E}\). The maximum torque will be obtained when the angle \(胃\) is \(90掳\), according to the property of the sine function. Therefore, for maximum torque, the dipole should be oriented perpendicular to the electric field. Here, as \(\vec{E}\) is directed to the left, the dipole moment \(\vec{p}\) should be vertically aligned, either upwards or downwards.
02

Calculate the electric potential energy for maximum torque

The potential energy of a dipole in an electric field is given by \(U = -pE \cos 胃\). To find the potential energy corresponding to the maximum torque orientation, substitute \(胃 = 90掳\) in the equation. With \(\cos 90掳 = 0\), the potential energy \(U\) will become zero.
03

Determine the orientation for zero torque and maximum potential energy

The torque will be zero when \(胃 = 0掳\) or \(胃 = 180掳\). In both cases, the dipole moment \(\vec{p}\) is aligned with \(\vec{E}\), with the same or opposite directions. Note that, the potential energy will be maximum at \(胃 = 180掳\), because the cosine function has a minimum value of -1 when \(胃 = 180掳\). Thus, the orientation for zero torque and maximum potential energy is when \(\vec{p}\) is antiparallel to \(\vec{E}\), in the opposite direction.
04

Define the type of equilibrium

The equilibrium is unstable. This is because in such a state, even a slight rotation of the dipole caused by disturbances will make it rotate until it is parallel to the electric field, contrary to the initial orientation. This characterizes an unstable equilibrium state in the presence of an electric field.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Electric Dipole Moment
The electric dipole moment is a vector quantity that measures the separation of positive and negative charges within an electric dipole. It is represented by the symbol \(\vec{p}\) and is defined as the product of the charge magnitude \(q\) and the separation distance \(d\) between the charges. Mathematically, \(\vec{p} = q \vec{d}\). The direction of the dipole moment is from the negative to the positive charge. In our exercise, the orientation of \(\vec{p}\) relative to an external electric field \(\vec{E}\) influences the behavior of the dipole, including the torque it experiences and its potential energy.
Torque on Electric Dipole
Torque is an important concept when analyzing the behavior of an electric dipole in an electric field. It is a measure of the force that causes the dipole to rotate, which is given by \(\tau = pE \sin \theta\). Here, \(\tau\) represents torque, \(p\) is the magnitude of the electric dipole moment, \(E\) is the magnitude of the electric field, and \(\theta\) is the angle between \(\vec{p}\) and \(\vec{E}\). The maximum torque occurs when \(\theta = 90^\circ\), meaning the dipole is perpendicular to the electric field. This maximum torque is critical in understanding the behavior of the dipole in various orientations relative to the field.
Electric Potential Energy
The electric potential energy of a dipole in an electric field describes the energy due to its position and orientation. It is determined using the formula \(U = -pE \cos \theta\), where \(U\) is the electric potential energy. When the orientation yields maximum torque (at \(\theta = 90^\circ\)), the \(\cos\) of \(\theta\) equals zero, resulting in zero potential energy. Conversely, when the torque is zero (\(\theta = 0^\circ\) or \(\theta = 180^\circ\)), the potential energy reaches its maximum value. The relationship between orientation and potential energy is vital to understanding the stability of an electric dipole in a field.
Equilibrium of Electric Dipole
An electric dipole can be in different states of equilibrium in an electric field. When the dipole is aligned with the field (either in the same or opposite direction), it experiences zero torque, with the potential energy being at a maximum or minimum. If the dipole is aligned in the opposite direction to \(\vec{E}\), it is in unstable equilibrium. A slight disturbance would cause the dipole to rotate towards alignment with the field, which represents a lower energy state. This type of equilibrium is crucial for understanding the motion and stability of molecules and materials in electric fields.

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Most popular questions from this chapter

If we rub a balloon on our hair, the balloon sticks to a wall or ceiling. This is because the rubbing transfers electrons from our hair to the balloon, giving it a net negative charge. When the balloon is placed near the ceiling, the extra electrons in it repel nearby electrons in the ceiling, creating a separation of charge on the ceiling, with positive charge closer to the balloon. Model the interaction as two point-like charges of equal magnitude and opposite signs, separated by a distance of \(500 \mu \mathrm{m}\). Neglect the more distant negative charges on the ceiling. (a) A typical balloon has a mass of \(4 \mathrm{~g}\). Estimate the minimum magnitude of charge the balloon requires to stay attached to the ceiling. (b) since a balloon sticks handily to the ceiling after being rubbed, assume that it has attained 10 times the estimated minimum charge. Estimate the number of electrons that were transferred to the balloon by the process of rubbing.

Three identical point charges \(q\) are placed at each of three corners of a square of side \(L\). Find the magnitude and direction of the net force on a point charge \(-3 q\) placed (a) at the center of the square and (b) at the vacant corner of the square. In each case, draw a free-body diagram showing the forces exerted on the \(-3 q\) charge by each of the other three charges.

Two thin rods, each with length \(L\) and total charge \(+Q,\) are parallel and separated by a distance \(a .\) The first rod has one end at the origin and its other end on the positive \(y\) -axis. The second rod has its lower end on the positive \(x\) -axis. (a) Explain why the \(y\) -component of the net force on the second rod vanishes. (b) Determine the \(x\) -component of the differential force \(d F_{2}\) exerted on a small portion of the second rod, with length \(d y_{2}\) and position \(y_{2},\) by the first rod. (This requires integrating over differential portions of the first rod, parameterized by \(\left.d y_{1} .\right)\) (c) Determine the net force \(\vec{F}_{2}\) on the second rod by integrating \(d F_{2 x}\) over the second rod. (d) Show that in the limit \(a \gg L\) the force determined in part (c) becomes \(\frac{1}{4 \pi \epsilon_{0}} \frac{Q^{2}}{a^{2}} \hat{\imath}\). (e) Determine the external work required to move the second rod from very far away to the position \(x=a\), provided the first rod is held fixed at \(x=0 .\) This describes the potential energy of the original configuration. (f) Suppose \(L=50.0 \mathrm{~cm}, a=10.0 \mathrm{~cm}, Q=10.0 \mu \mathrm{C},\) and \(m=500 \mathrm{~g}\). If the two rods are released from the original configuration, they will fly apart and ultimately achieve a particular relative speed. What is that relative speed?

Two small spheres spaced \(20.0 \mathrm{~cm}\) apart have equal charge. How many excess electrons must be present on each sphere if the magnitude of the force of repulsion between them is \(3.33 \times 10^{-21} \mathrm{~N} ?\)

A proton is traveling horizontally to the right at \(4.50 \times 10^{6} \mathrm{~m} / \mathrm{s}\). (a) Find the magnitude and direction of the weakest electric field that can bring the proton uniformly to rest over a distance of \(3.20 \mathrm{~cm}\). (b) How much time does it take the proton to stop after entering the field? (c) What minimum field (magnitude and direction) would be needed to stop an electron under the conditions of part (a)?
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