A thin disk with a circular hole at its center, called an anmulus, has inner radius $R_1$ and outer radius $R_2$. The disk has a uniform positive surface charge density $\sigma$ on its surface. (c) Show that at points on the $x$-axis that are sufficiently close to the origin, the magnitude of the electric field is approximately proportional to the distance between the center of the annulus and the point. How close is "sufficiently close"?

A thin, flat, uniform disk has mass $M$ and radius $R$. A circular hole of radius $R / 4$, centered at a point $R / 2$ from the disk's center, is then punched in the disk. $(a)$ Find the moment of inertia of the disk with the hole about an axis through the original center of the disk, perpendicular to the plane of the disk. (Hint: Find the moment of inertia of the piece punched from the disk.) $(b)$ Find the moment of inertia of the disk with the hole about an axis through the center of the hole, perpendicular to the plane of the disk.

The ammonia molecule (NH ) has a dipole moment of $5.0 \times 10^{-30} \mathrm{C} \cdot \mathrm{m}$. Ammonia molecules in the gas phase are placed in a uniform electric field $\vec{E}$ with magnitude $1.6 \times 10^{6} \mathrm{N} / \mathrm{C}$.

(a) What is the change in electric potential energy when the dipole moment of a molecule changes its orientation with respect to E S from parallel to perpendicular?

(b) At what absolute temperature T is the average translational kinetic energy $\frac{3}{2} k T$ of a molecule equal to the change in potential energy calculated in part (a)? (Note: Above this temperature, thermal agitation prevents the dipoles from aligning with the electric field.)

Starting from rest, a disk rotates about its central axis with constant angular acceleration. In 5.0 s, it rotates 20 rad. During that time, what are the magnitudes of (a) the angular acceleration and (b) the average angular velocity? (c) What is the instantaneous angular velocity of the disk at the end of the 5.0 s? (d) With the angular acceleration unchanged, through what additional angle will the disk turn during the next 5.0 s?

The circular disk rotates about an axis through its center O and has three holes of diameter d positioned as shown. A fourth hole is to be drilled in the disk at the same radius r so that the disk will be in balance (mass center at O). Determine the required diameter D of the new hole and its angular position.

A disk of radius R has an initial mass M. Then a hole of radius R / 4 is drilled, with its edge at the disk center as shown in figure. Find the new rotational inertia about the central axis. (Hint: Find the rotational inertia of the missing piece, and subtract it from that of the whole disk.)

An annulus (a disk with a hole) made of paper has an outer radius $R$ and an inner radius $R / 2$. An amount $Q$ of electric charge is uniformly distributed over the paper. $(a)$ Find the potential as a function of distance on the axis of the annulus. $(b)$ Find the electric field on the axis of the annulus.

The ammonia molecule $\left(\mathrm{NH}_3\right)$ has a dipole moment of $5.0 \times 10^{-30} \mathrm{C} \cdot \mathrm{m}$. Ammonia molecules in the gas phase are placed in a uniform electric field $\overrightarrow{\boldsymbol{E}}$ with magnitude $1.6 \times$ $10^6 \mathrm{~N} / \mathrm{C}$. (b) at what absolute temperature $t$ is the average translational kinetic energy $\frac{3}{2} k t$ of a molecule equal to the change in potential energy calculated in part (a)? (note: above this temperature, thermal agitation prevents the dipoles from aligning with the electric field.)

The ammonia molecule

$$NH_3$$

has a permanent electric dipole moment equal to 1.47 D, where 1 D = 1 debye unit =

$$3.34 × 10^-30 C·m$$

. Calculate the electric potential due to an ammonia molecule at a point 52.0 nm away along the axis of the dipole. (Set V = 0 at infinity.)

The ammonia molecule

$$NH_3$$

has a permanent electric dipole moment equal to 1.47 D, where

$$1 D = 1\ debye\ unit\ = 3.34 \times 10 ^ { - 30 } \mathrm { C } \cdot \mathrm { m }$$

. Calculate the electric potential due to an ammonia molecule at a point 103 nm away along the axis of the dipole. (Set V = 0 at infinity.)

An ammonia molecule $\left(\mathrm{NH}_{3}\right)$ has a permanent electric dipole moment $5.0 \times 10^{-30} \mathrm{Cm}$. A proton is 2.0 nm from the molecule in the plane that bisects the dipole. What is the electric force of the molecule on the proton?

An electron is projected with an initial speed $v_0=$ $1.60 \times 10^6 \mathrm{~m} / \mathrm{s}$ into the uniform field between two parallel plates, Assume that the field between the plates is uniform and directed vertically downward and that the field outside the plates is zero. The electron enters the field at a point midway between the plates. (b) Suppose that the electron in Fig. we saw earlier is replaced by a proton with the same initial speed $v_0$. Would the proton hit one of the plates? If not, what would be the magnitude and direction of its vertical displacement as it exits the region between the plates?

An electron is projected with an initial speed of

$$1.6 \times 10 ^ { 5 } \mathrm { m } / \mathrm { s }$$

directly toward a proton that is fixed in place. If the electron is initially a great distance from the proton, at what distance from the proton is the speed of the electron instantaneously equal to twice the initial value?

An electron is projected with an initial speed $v_0=$ $1.60 \times 10^6 \mathrm{~m} / \mathrm{s}$ into the uniform field between two parallel plates, Assume that the field between the plates is uniform and directed vertically downward and that the field outside the plates is zero. The electron enters the field at a point midway between the plates.(d) Discuss whether it is reasonable to ignore the effects of gravity for each particle.