Electricity and Magnetism
# Magnetic Fields

**in Tesla** of the magnetic field in the middle of this loop of wire?

$E$ and a uniform vertical (out of the page) magnetic $B$ field inside cyclotron which is perpendicular to the electric field. The electric field switches such that is always points toward the dee the particle is not in.

Consider a standard cyclotron accelerator in which two semi-circular regions (dees) are connected to an AC voltage source which provides an electric fieldA proton is released from rest such that it starts rotating in the cyclotron at radius $R$ and finally comes out from the slits of the cyclotron. The distance between the two semi-circular regions is $d$. Find the **maximum** number of turns proton take before coming out from slit's.

**Details and Assumptions**

${ { m }_{ p }=1.6\times { 10 }^{ -27 }\text{ kg}\\ { q }_{ p }=1.6\times { 10 }^{ -19 } C\\ B={ 10 }^{ -4 }\text{ T}\\ R=6\text{ m}\\ E=10\frac{\text{V}}{\text{m}}\\ d=10\text{ cm}\\ }$.

The charged particles from solar eruptions hit Earth mainly around the North and South poles (and cause auroras). This is because our Earth is similar to a big magnet. The Earth generates a magnetic field and this magnetic field funnels the charged particles towards the poles. In order to see an example of this funneling, we can think of the following problem:

Consider each magnetic pole separately (so there's only one pole in the problem). The magnetic field near a single pole is $\vec{B} = k\vec{r}/r^3=k\hat{r}/r^2$ where $\vec{r}$ is the radial vector point of the pole and the point of interest and $\hat{r}=\vec{r}/r$ is the radial unit vector. The sign of $k$ is opposite for the North and South poles. If there's an electric charge moving in that magnetic field, its trajectory is on a surface of a cone, i.e. a big funnel. Find the vertex angle (the angle between the axis and a line on the surface of the cone) **in degrees** with these given initial conditions: the distance between the charge and the pole is $r = 1~\mbox{m}$ and the velocity vector of the charge is $v = 2~\mbox{m/s}$ perpendicular to the line connecting the pole and the charge. We will consider a north pole and so let $k = 3~\mbox{T}\cdot\mbox{m}^2$. The charge of the particle is $q = 4~\mbox{C}$ and the mass is $m = 5~\mbox{kg}$.

**Details and assumptions**

- Hint: For anyone who has not taken electromagnetism yet, the force on the charge particle is given by the Lorentz force law: $\vec{F}=q\vec{v} \times \vec {B}$.

$E=120 ~\textrm{kV/m}$) and magnetic ($B=50 ~\textrm{mT}$) fields. Then the beam strikes a grounded target. Find the force **in Newtons** with which the beam acts on the target if the beam current is $I=0.8 ~ \textrm{mA}$.

Assume that the collisions with the target are inelastic.

The proton's mass and charge are:
$m_{p}=1.67 \times 10^{-27}\textrm{ kg}$
$e= 1.6 \times 10^{-19} C.$