Electricity and Magnetism
# E+M Warmups

Consider a small black ball of radius $R$ and density $\rho= 1~\frac{\mbox{g}}{\mbox{cm}^{3}}$ located at a certain distance above the surface of the Sun. For what radius $R$ **in micrometers** is the gravitational attraction of the Sun counterbalanced by the radiation force?

Assume that the black ball absorbs all the incident light and that the total power radiated by the Sun is $P=4\times 10^{26}~\mbox{W}$.

**Details and assumptions**

The mass of the Sun is $M_{s}=2\times 10^{30}~\mbox{kg}$. The universal constant of gravitation is $G=6.67\times 10^{-11}~\text{m}^{3}\text{kg}^{-1} \text{s}^{-2}$ and the speed of light $c=3\times 10^{8}~\mbox{m/s}$.

Two 1-g beads with charges $Q_{x}=2~\mu\mbox{C}$ and $Q_{y}=-3~\mu\mbox{C}$ slide on (without friction) two perpendicular wires ($x$ and $y$ axes). The charges are initially located at $(5~\mbox{m},0)$ and $(0,3~\mbox{m})$ and released from rest.

What is the distance **in meters** between the charges when $Q_{y}$ crosses the origin $(0,0)$?

**Details and assumptions**

$k=\frac{1}{4\pi \epsilon_{0}}=9 \times 10^{9} \mbox{N m}^{2}/\mbox{C}^{2}$

$q_{A}=0$ and the spheres at B and C carry the same charge $q_{B}=q_{C}=q.$ It is known that the sphere B exerts an electrostatic force on C which has a magnitude $F=4~\mbox{N}.$ Suppose an engineer connects a very thin conducting wire between spheres A and B. Then she removes the wire and connects it between spheres A and C. After these operations, what is the magnitude of the force of interaction **in Newtons** between B and C?

The Hyperloop is a hypothetical new fast transport system between cities, which works by launching pods that carry people through a very low air pressure tunnel. If we model the Hyperloop pod as a hollow metal cylinder with a radius of $1~\mbox{m}$ and a length of $10~\mbox{m}$, what is the maximum voltage difference between the sides of the pod **in Volts**?

**Details and assumptions**

- The magnetic field at the earth's surface is $0.5~\mbox{Gauss}$.
- The speed of the pod is $300~\mbox{m/s}$.