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Chapter 23: Problem 7

Which of the following angles between an electric dipole moment and an applied electric field will result in the most stable state? a) \(0 \mathrm{rad}\) d) The electric dipole moment is b) \(\pi / 2\) rad not stable under any condition in c) \(\pi\) rad an applied electric field.

Short Answer

Expert verified
Answer: a) \(0 \mathrm{rad}\)

Step by step solution

01

Understand the electric dipole moment and potential energy

In a system with an electric dipole moment and an applied electric field, the potential energy (U) of the system can be described as: $$ U = -\vec{p} \cdot \vec{E} $$ where \(\vec{p}\) is the electric dipole moment and \(\vec{E}\) is the applied electric field. The dot product between the two vectors depends on the angle between them, denoted as \(\theta\). Therefore, the potential energy formula can be rewritten as: $$ U = -p E \cos{\theta} $$
02

Analyze the potential energy with respect to the angle θ

We need to determine which angle θ between the electric dipole moment and applied electric field results in the most stable state. The most stable state corresponds to the lowest potential energy. We can examine the given options one by one. Let's evaluate the potential energy for each given angle: 1. Angle θ = \(0 \mathrm{rad}\): $$ U_{0} = -p E \cos{(0 \mathrm{rad})} = -p E $$ 2. Angle θ = \(\pi / 2\) rad: $$ U_{\pi/2} = -p E \cos{(\pi / 2 \mathrm{rad})} = 0 $$ 3. Angle θ = \(\pi\) rad: $$ U_{\pi} = -p E \cos{(\pi \mathrm{rad})} = p E $$
03

Determine the angle corresponding to the most stable state

To find the most stable state, we need to find the angle that results in the lowest potential energy. Comparing the values of the potential energy calculated in step 2, we can see that: - \(U_{0}\) = \(-p E\): The potential energy is negative, indicating a stable state. - \(U_{\pi/2}\) = \(0\): The potential energy is equal to zero, indicating a less stable state than when the potential energy is negative. - \(U_{\pi}\) = \(p E\): The potential energy is positive, indicating an unstable state. Therefore, the angle that results in the most stable state is \(0 \mathrm{rad}\), as it leads to the lowest potential energy. Hence, the correct answer is: a) \(0 \mathrm{rad}\)

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Most popular questions from this chapter

A solid conducting sphere of radius \(R\) is centered about the origin of an \(x y z\) -coordinate system. A total charge \(Q\) is distributed uniformly on the surface of the sphere. Assuming, as usual, that the electric potential is zero at an infinite distance, what is the electric potential at the center of the conducting sphere? a) zero c) \(Q / 2 \pi \epsilon_{0} R\) b) \(Q / \epsilon_{0} R\) d) \(Q / 4 \pi \epsilon_{0} R\)

Use \(V=\frac{k q}{r}, E_{x}=-\frac{\partial V}{\partial x}, E_{y}=-\frac{\partial V}{\partial y},\) and \(E_{z}=-\frac{\partial V}{\partial z}\) to derive the expression for the electric field of a point charge, \(q\)

A total charge of \(Q=4.2 \cdot 10^{-6} \mathrm{C}\) is placed on a conducting sphere (sphere 1 ) of radius \(R=0.40 \mathrm{~m}\). a) What is the electric potential, \(V_{1},\) at the surface of sphere 1 assuming that the potential infinitely far away from it is zero? (Hint: What is the change in potential if a charge is brought from infinitely far away, where \(V(\infty)=0,\) to the surface of the sphere?) b) A second conducting sphere (sphere 2) of radius \(r=0.10 \mathrm{~m}\) with an initial net charge of zero \((q=0)\) is connected to sphere 1 using a long thin metal wire. How much charge flows from sphere 1 to sphere 2 to bring them into equilibrium? What are the electric fields at the surfaces of the two spheres?

In molecules of gaseous sodium chloride, the chloride ion has one more electron than proton, and the sodium ion has one more proton than electron. These ions are separated by about \(0.24 \mathrm{nm}\). How much work would be required to increase the distance between these ions to \(1.0 \mathrm{~cm} ?\)

A negatively charged particle revolves in a clockwise direction around a positively charged sphere. The work done on the negatively charged particle by the electric field of the sphere is a) positive. b) negative. c) zero.
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