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.

The identical small spheres shown in FIGURE P22.64 are charged to +100 nC and −100 nC. They hang as shown in a 100,000 N/C electric field. What is the mass of each sphere?

A 10.0 nC charge is located at position (x, y)=(1.0 cm, 2.0 cm). At what (x, y) position(s) is the electric field $(21,600\hat{i}-28,800\hat{j})$ N/C?

Three 1.0 nC charges are placed as shown in FIGURE P22.66. Each of these charges creates an electric field E at a point 3.0 cm in front of the middle charge. What are the three fields E₁, E₂, and E₃ created by the three charges? Write your answer for each as a vector in component form.

An electric field $\overrightarrow{E}=200,000\hat{i}$ N/C causes the point charge in FIGURE P22.68 to hang at an angle. What is θ?

Starting from rest, how long does it take an electron to move 1.0 cm in a steady electric field of magnitude 100 N/C?

A 5.0 g ball charged to 1.5 μC is tied to a 25-cm-long string. It swings at 250 rpm in a horizontal circle around a stationary ball charged to −2.5 μC. What is the tension in the string?

Space explorers discover an 8.7×10¹⁷ kg asteroid that happens to have a positive charge of 4400 C. They would like to place their 3.3×10⁵ kg spaceship in orbit around the asteroid. Interestingly, the solar wind has given their spaceship a charge of −1.2 C. What speed must their spaceship have to achieve a 7500-km-diameter circular orbit?

A small 1.0 g block charged to 75 nC is placed on a 30° inclined plane. The coefficients of static and kinetic friction are 0.20 and 0.10, respectively. What minimum strength horizontal electric field is needed to keep the block from sliding down the plane?

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 $F_{net}=2KqQs/r^3$.