24. Electric Force & Field; Gauss' Law

An electric dipole with dipole moment $p$ is in a uniform external electric field $E$. Show that for the stable orientation in part (b), the dipole's own electric field tends to oppose the external field. Note: Part (b) asked 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.) Also, part (a) asked to find the orientations of the dipole for which the torque on the dipole is zero.

An electric dipole with dipole moment $p$ is in a uniform external electric field $E$. 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.) Note: Part (a) asked to find the orientations of the dipole for which the torque on the dipole is zero.

An electric dipole with dipole moment $p$ is in a uniform external electric field $E$. Find the orientations of the dipole for which the torque on the dipole is zero.

Point charges $q_1=-4.5$ nC and $q_2=+4.5$ nC are separated by $3.1$ mm, forming an electric dipole. The charges are in a uniform electric field whose direction makes an angle of $36.9$° with the line connecting the charges. What is the magnitude of this field if the torque exerted on the dipole has magnitude $7.2\times {10}^{-9}$ Nm?

The dipole moment of the water molecule (H₂O) is $6.17\times {10}^{-30}$ Cm. Consider a water molecule located at the origin whose dipole moment $p$ points in the $+x$-direction. A chlorine ion (Cl^-), of charge $-1.60\times {10}^{-19}$ C, is located at $x=3.00\times {10}^{-9}$ m. Find the magnitude and direction of the electric force that the water molecule exerts on the chlorine ion. Is this force attractive or repulsive? Assume that $x$ is much larger than the separation $d$ between the charges in the dipole, so that the approximate expression for the electric field along the dipole axis derived in Example $21.14$ can be used.

An electric dipole consists of two opposite charges $\pm q$ separated by a small distance $s$. The product $p=qs$ is called the dipole moment. Figure P$22.61$ shows an electric dipole perpendicular to an electric field $E$. Find an expression in terms of $p$ and $E$ for the magnitude of the torque that the electric field exerts on the dipole.

In Section 22.3 we claimed that a charged object exerts a net attractive force on an electric dipole. Let's investigate this. FIGURE CP22.80 shows a permanent electric dipole consisting of charges +q and −q separated by the fixed distance s. Charge +Q is distance r from the center of the dipole. We'll assume, as is usually the case in practice, that s≪r. Use the binomial approximation $(1+x{)}^{-n}\approx 1-nx$ if x≪1 to show that your expression from part a can be written $F_{net}=2KqQs/r^3$.

An electric dipole is formed from two charges, ±q, spaced 1.0 cm apart. The dipole is at the origin, oriented along the y-axis. The electric field strength at the point (x, y)=(0 cm, 10 cm) is 360 N/C. What is the charge q? Give your answer in nC.

The permanent electric dipole moment of the water molecule (H₂O) is $6.2\times {10}^{-30}$ Cm. What is the maximum possible torque on a water molecule in a $5.0\times {10}^8$ N/C electric field?

Derive Equation 23.11 for the field Ē dipole in the plane that bisects an electric dipole.

An electric field can induce an electric dipole in a neutral atom or molecule by pushing the positive and negative charges in opposite directions. The dipole moment of an induced dipole is directly proportional to the electric field. That is, $\overrightarrow{p}=\alpha \overrightarrow{E}$, where α is called the polarizability of the molecule. A bigger field stretches the molecule farther and causes a larger dipole moment. An ion with charge q is distance r from a molecule with polarizability α. Find an expression for the force _{$\overrightarrow{E}$ion on dipole}.

FIGURE EX25.10 shows the potential energy of an electric dipole. Consider a dipole that oscillates between ±60°. What is the dipole's mechanical energy?

The HCl molecule has a dipole moment of about 3.4 x 10^-30 Cm. The two atoms are separated by about 1.0 x 10^-10 m. (a) What is the net charge on each atom? (b) Is this equal to an integral multiple of e? If not, explain. (c) What maximum torque would this dipole experience in a 2.5 x 10⁴ N/C electric field? (d) How much energy would be needed to rotate one molecule 45° from its equilibrium position of lowest potential energy?

The electric field strength 1.5 cm from an electric dipole, on the axis of the dipole, is 1.5×10⁵ N/C. If the dipole is replaced by a single charge, what magnitude charge in nC will give the same field strength 1.5 cm away?

(II) The HCl molecule has a dipole moment of about 3.4 x 10^-30 Cm. The two atoms are separated by about 1.0 x 10^-10 m. (c) What maximum torque would this dipole experience in a 2.5 x 10⁴ N/C electric field? (d) How much energy would be needed to rotate one molecule 45° from its equilibrium position of lowest potential energy?