In experiments, physicists routinely accelerate protons to speeds quite close to the speed of light. The mass of a proton is $1.67 \times 10^{-27} \mathrm{~kg}$, and the proton is moving with a speed of $0.99 \mathrm{c}$. (a) Calculate the proton's momentum, according to Newton's definition. (b) Calculate the proton's relativistic momentum. (c) Determine the ratio of the relativistic momentum to the Newtonian momentum.

A spherical water droplet of radius $25 \mu \mathrm{m}$ carries an excess 250 electrons. What vertical electric field is needed to balance the gravitational force on the droplet at the surface of the earth? 94. A proton enters the uniform electric field produced by the two charged plates shown below. The magnitude of the electric field is $4.0 \times 10^5 \mathrm{~N} / \mathrm{C}$, and the speed of the proton when it enters is $1.5 \times 10^7 \mathrm{~m} / \mathrm{s}$. What distance $d$ has the proton been deflected downward when it leaves the plates?

A proton, traveling with a velocity of $4.5 \times 10^{6} \mathrm{m} / \mathrm{s}$ due east, experiences a magnetic force that has a maximum magnitude of $8.0 \times 10^{-14} \mathrm{N}$ and a direction of due south. What are the magnitude and direction of the magnetic field causing the force? (b) Repeat part (a) assuming the proton is replaced by an electron.

At a particular instant, a proton at the origin has velocity $\left\langle 2 \times 10^{4},-2 \times 10^{4}, 0\right\rangle \mathrm{m} / \mathrm{s}$. You need to calculate the magnetic field at location $\langle 0.02,0.06,0\rangle$ m due to the moving proton. What is $\vec{v} \times \hat{r}?$

A proton (mass $m_{p}$), a deuteron $(m=2 m_{p}, Q=e)$ and an alpha particle $(m=4 m_{\mathrm{p}}, Q=2 e)$ are accelerated by the same potential difference V and then enter a uniform magnetic field $\overrightarrow{\mathbf{B}}$, where they move in circular paths perpendicular to $\overrightarrow{\mathbf{B}}$. Determine the radius of the paths for the deuteron and alpha particle in terms of that for the proton.

Two photons are produced when a proton and an antiproton annihilate each other. What is the minimum frequency and corresponding wavelength of each photon?

Complete the table.

A proton is located at $\left\langle 0,0,-2 \times 10^{-9}\right\rangle \mathrm{m},$ and an alpha particle (consisting of two protons and two neutrons) is located at $\left\langle 1.5 \times 10^{-9}, 0,2 \times 10^{-9}\right\rangle \mathrm{m}.$ Calculate the force the alpha particle exerts on the proton.

Assume that a ball of charged particles has a uniformly distributed negative charge density except for a narrow radial tunnel through its center, from the surface on one side to the surface on the opposite side. Also assume that we can position a proton anywhere along the tunnel or outside the ball. Let

$$F_R$$

be the magnitude of the electrostatic force on the proton when it is located at the ball's surface, at radius R. As a multiple of R, how far from the surface is there a point where the force magnitude is

$$0.70F_R$$

if we move the proton (a) away from the ball and (b) into the tunnel?

Assume that a ball of charged particles has a uniformly distributed negative charge density except for a narrow radial tunnel through its center, from the surface on one side to the surface on the opposite side. Also assume that we can position a proton anywhere along the tunnel or outside the ball. Let $F_R$ be the magnitude of the electrostatic force on the proton when it is located at the ball’s surface, at radius R. As a multiple of R, how far from the surface is there a point where the force magnitude is 0.50 $F_R$ if we move the proton away from the ball?

A proton is located at $\left\langle 0,0,-2 \times 10^{-9}\right\rangle\ \mathrm{m},$ and an alpha particle (consisting of two protons and two neutrons) is located at $\left\langle 1.5 \times 10^{-9}, 0,2 \times 10^{-9}\right\rangle\ \mathrm{m} .$ Calculate the force the proton exerts on the alpha particle.

A proton $\left(q=1.6 \times 10^{-19} \mathrm{C} ; m=1.7 \times 10^{-27} \mathrm{kg}\right)$ is in a uniform 0.25 T magnetic field. The proton moves in a clockwise circle with a tangential speed of $2.8 \times 10^{5}$ m/s. What is the direction of the magnetic field? Explain how you determined this.

A wire lying along the $x$ axis carries a current of $30 \mathrm{~A}$ in the $+x$ direction. A proton at $\mathbf{r}=2.5 \mathrm{j}$ has instantaneous velocity $\mathbf{v}=2.0 \mathbf{i}-3.0 \mathbf{j}+4.0 \mathbf{k}$, where $\mathbf{r}$ is in meters and $\mathbf{v}$ is in meters per second. What is the instantaneous magnetic force on this proton?

At a particular instant, a proton is traveling west to east with a kinetic energy of $10 \mathrm{keV}$. Earth's magnetic field has a horizontal component of $1.8 \times 10^{-5} \mathrm{~T}$ north and a vertical component of $5.0 \times 10^{-5} \mathrm{~T}$ down. $(a)$ What is the path of the proton? $(b)$ What is the radius of curvature of the path?