Electricity and Magnetism
# E+M Warmups

**paraxial** rays as shown in the figure below. If $g=0.33~\mbox{mm}^{-1}$, determine the distance $d$ **in millimeters** between two consecutive focal points.

**in Watts** must develop the external forces to keep the ring rotating at constant angular speed $\omega$ (on average).

**in meters per second squared**? The radius of the balls is $R=1~\text{m}$.

An electron is accelerated by a potential difference of $U_e= 1~\mbox{mV}$. It then enters a region with an inhomogeneous magnetic field $\vec{B}(x,y,z)$ generated by a system of coils carrying currents $I_{1}, I_{2}\ldots$.

We then perform a similar experiment with a proton. We first reverse the current in all the coils generating the magnetic field. We then accelerate the proton with a potential difference of $U_p$ and it enters the region with the magnetic field. What must $U_p$ be **in Volts** so that the trajectory of the proton is the same as that of the electron? Don't forget any sign changes!

Hint: Compute $\frac{d}{ds} \frac{\vec{v}}{v},$ where $s$ is the arc length along the trajectory:

$s=\int_0^t v(t') dt'$.

**Details and assumptions**

The proton to electron mass ratio is approximately $6 \pi^{5}$.

In 1-d Newtonian mechanics the energy as a function of momentum for a particle is given by $E=\frac{1}{2}mv^2$ or, in terms of momentum, $E=p^2/(2m)$. In Einstein's theory of special relativity there is still energy and momentum, and they are still conserved, but the relationship between the two is different. In empty space, the energy of a particle as a function of its momentum $\vec{p}$ and mass $m$ is given by $E^2=c^2 \vec{p} \cdot \vec{p}+m^2c^4$ where $c$ is the speed of light. The photon, the "particle" of light, has no mass - its energy satisfies $E=c|\vec{p}|$ in empty space. In a medium like water this can change - photons travel slower in a medium and their energy-momentum relation becomes $E=c_m |\vec{p}|$, where $c_m$ is the speed of light in the medium. Our question is the following: A very high energy proton with $E_{prot}^2=c^2 \vec{p} \cdot \vec{p}+m_{prot}^2c^4$ enters a tank of water. What is the minimum magnitude of the proton's momentum, i.e. $|\vec{p}|$, in **kg m/s**, such that the proton can emit a photon with non-zero momentum and still conserve energy and momentum?

**Details and assumptions**

- The speed of light in empty space is $3 \times 10^8~\mbox{m/s}$.
- The mass of the proton is $1.67 \times 10^{-27}~\mbox{kg}$.
- The speed of light in water is $2.3 \times 10^8~\mbox{m/s}$.
- When photon emission begins to occur the initial proton, the final proton, and the emitted photon all travel in the same direction.