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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

Neurons are components of the nervous system of the body that transmit signals as electrical impulses travel along their length. These impulses propagate when charge suddenly rushes into and then out of a part of the neuron called an axon. Measurements have shown that, during the inflow part of this cycle, approximately \(5.6 \times 10^{11} \mathrm{Na}^{+}\) (sodium ions) per meter, each with charge \(+e,\) enter the axon. How many coulombs of charge enter a \(1.5 \mathrm{~cm}\) length of the axon during this process?

Two very large parallel sheets are \(5.00 \mathrm{~cm}\) apart. Sheet \(A\) carries a uniform surface charge density of \(-8.80 \mu \mathrm{C} / \mathrm{m}^{2},\) and sheet \(B,\) which is to the right of \(A,\) carries a uniform charge density of \(-11.6 \mu \mathrm{C} / \mathrm{m}^{2} .\) Assume that the sheets are large enough to be treated as infinite. Find the magnitude and direction of the net electric field these sheets produce at a point (a) \(4.00 \mathrm{~cm}\) to the right of sheet \(A ;\) (b) \(4.00 \mathrm{~cm}\) to the left of sheet \(A ;\) (c) \(4.00 \mathrm{~cm}\) to the right of sheet \(B\).

Two point charges are placed on the \(x\) -axis as follows: Charge \(q_{1}=+4.00 \mathrm{nC}\) is located at \(x=0.200 \mathrm{~m},\) and charge \(q_{2}=+5.00 \mathrm{nC}\) is at \(x=-0.300 \mathrm{~m} .\) What are the magnitude and direction of the total force exerted by these two charges on a negative point charge \(q_{3}=-6.00 \mathrm{nC}\) that is placed at the origin?

Lightning occurs when there is a flow of electric charge (principally electrons) between the ground and a thundercloud. The maximum rate of charge flow in a lightning bolt is about \(20,000 \mathrm{C} / \mathrm{s} ;\) this lasts for \(100 \mu \mathrm{s}\) or less. How much charge flows between the ground and the cloud in this time? How many electrons flow during this time?

Point charge \(A\) is on the \(x\) -axis at \(x=-3.00 \mathrm{~cm}\). At \(x=1.00 \mathrm{~cm}\) on the \(x\) -axis its electric field is \(2700 \mathrm{~N} / \mathrm{C}\). Point charge \(B\) is also on the \(x\) -axis, at \(x=5.00 \mathrm{~cm}\). The absolute magnitude of charge \(B\) is twice that of \(A .\) Find the magnitude and direction of the total electric field at the origin if (a) both \(A\) and \(B\) are positive; (b) both are negative; (c) \(A\) is positive and \(B\) is negative; (d) \(A\) is negative and \(B\) is positive.
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