Show that the magnetic field strength between the plates a distance $r$ from the axis through the centers of both plates is given by $B=\left(1.9 \times 10^{-3} \mathrm{~T} / \mathrm{m}\right) r$.

Some science fiction writers have described solar sails that could propel interstellar spaceships. Imagine a giant sail on a spacecraft subjected to radiation pressure from our Sun. Using a reasonable value for A, compare the force on the spacecraft due to the radiation pressure and the force on the spacecrait due to the gravitational force of the Sun on the spacecraft. Does the result imply that such a system will work?

Some science fiction writers have described solar sails that could propel interstellar spaceships. Imagine a giant sail on a spacecraft subjected to radiation pressure from our Sun. If the sail is assumed highly reflective, show that the force exerted by the sunlight on the spacecraft's sail is given by $P_5 A /\left(4 \pi r^2 c\right)$ where $P_5$ is the power output of the Sun $\left(3.8 \times 10^{26} \mathrm{~W}\right), A$ is the surface area of the sail, $r$ is the distance from the Sun, and $c$ is the speed of light. (Assume the area of the sail is much larger than the area of the spacecraft so that all the force is due to radiation pressure on the sail only.)

Some science fiction writers have described solar sails that could propel interstellar spaceships. Imagine a giant sail on a spacecraft subjected to radiation pressure from our Sun. Determine this arrangement works better if the sail is highly reflective rather than highly absorptive.

A circuit consists of a coil that has a resistance equal to $8.00 \Omega$ and a self-inductance equal to $4.00 \mathrm{mH}$, an open switch, and an ideal $100-\mathrm{V}$ battery-all connected in series. At $t=0$ the switch is closed. Find the current and its rate of change at times $(a) t=0$, (b) $t=0.100 \mathrm{~ms},(c) t=0.500 \mathrm{~ms}$, and $(d) t=1.00 \mathrm{~ms}$. Find the magnetic field strength and magnetic energy density at the mean radius.

A $100$-turn circular coil has a diameter of $2.00 \mathrm{~cm}$ and a resistance of $50.0\ \Omega$, and the two ends of the coil are connected together. The plane of the coil is perpendicular to a uniform magnetic field of magnitude $1.00 \mathrm{~T}$. The direction of the field is reversed. Determine the total charge that passes through a cross-section of the wire. If the reversal takes $0.100 \mathrm{~s}$, determine

Determine the magnetic field at point $P$ in Figure 27-60.

One of the first successful satellites launched by the United States in the 1950 s was essentially a large spherical (aluminized) Mylar balloon from which radio signals were reflected. After several orbits around Earth, scientists noticed that the orbit itself was changing with time. They eventually determined that radiation pressure from sunlight was causing the orbit of this object to change- a phenomenon not taken into account in planning the mission. Find the ratio of the radiation-pressure force by the sunlight on the satellite to the gravitational force by Earth's gravity on the satellite.

A proton has a velocity of $1.0 \times 10^2 \mathrm{~m} / \mathrm{s} \hat{i}+2.0 \times$ $10^2 \mathrm{~m} / \mathrm{s} \hat{j}$ and is located in the $z=0$ plane at $x=3.0 \mathrm{~m}, y=4.0 \mathrm{~m}$ at some time $t=T$. Determine the magnetic field in the $z=0$ plane at $

A proton has a velocity of $1.0 \times 10^2 \mathrm{~m} / \mathrm{s} \hat{i}+2.0 \times$ $10^2 \mathrm{~m} / \mathrm{s} \hat{j}$ and is located in the $z=0$ plane at $x=3.0 \mathrm{~m}, y=4.0 \mathrm{~m}$ at some time $t=T$. Determine the magnetic field in the $z=0$ plane at $x=3.0 \mathrm{~m}, y=6.0 \mathrm{~m}$.

A small current element at the origin has a length of $2.0 \mathrm{~mm}$ and carries a current of $2.0 \mathrm{~A}$ in the $+z$ direction. Determine the magnitude of the magnetic field due to this current element and indicate its direction on a diagram at $x=2.0 \mathrm{~m}, y=4.0 \mathrm{~m}, z=0$.

Suppose that in Figure 26-35, the magnetic field has a magnitude of $60\ \mathrm{mT}$, the distance $d$ is $40 \mathrm{~cm}$, and $\theta$ is $24^{\circ}$. Determine the speed $v$ at which a particle enters the region and the exit angle $\phi$ if the particle is a proton.

In laser cooling and trapping, the forces associated with radiation pressure are used to slow down atoms from thermal speeds of hundreds of meters per second at room temperature to speeds of a few meters per second or slower. An isolated atom will absorb only radiation of specific frequencies. If the frequency of the laser-beam radiation is tuned so that the target atoms will absorb the radiation, then the radiation is absorbed during a process called resonant absorption. The cross-sectional area of the atom for resonant absorption is approximately equal to $\lambda^2$, where $\lambda$ is the wavelength of the laser light. About how long would it take such a light beam to slow a rubidium atom in a gas at room temperature $(300 \mathrm{~K})$ to near-zero speed?

In laser cooling and trapping, the forces associated with radiation pressure are used to slow down atoms from thermal speeds of hundreds of meters per second at room temperature to speeds of a few meters per second or slower. An isolated atom will absorb only radiation of specific frequencies. If the frequency of the laser-beam radiation is tuned so that the target atoms will absorb the radiation, then the radiation is absorbed during a process called resonant absorption. The cross-sectional area of the atom for resonant absorption is approximately equal to $\lambda^2$, where $\lambda$ is the wavelength of the laser light. Estimate the acceleration of a rubidium atom (molar mass $85 \mathrm{~g} / \mathrm{mol})$ in a laser beam whose wavelength is $780 \mathrm{~nm}$ and intensity is $10 \mathrm{~W} / \mathrm{m}^2$.