Physik M
07.12.2017

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Problem 1

A proton ($m=1.6726\times10^{-27}$ kg, $q=1.6022\times10^{-19}$ C) moves in a region where there is an electric field and a magnetic field,

At time $t=0$, the proton is at $\vec{r}=0$ and has a velocity of $\vec{v}=$  $\hat{x}$ m/s. Where is the proton at time $t=1$ s?

$\vec{r} = $ $\,\hat{x}+$$\,\hat{y}+$$\,\hat{z}$ [m]

Problem 2

A current of $I=$  [A] flows through a square wire. The wire lies in the $z = 0$ plane. The direction of the current is given by the arrow. All sides of the square are 20 cm long.

What is the magnetic field at point $O$ in the middle of the wire?

$\vec{B} = $ $\hat{x} + $ $\hat{y} + $ $\hat{z}$ [T]

What is the unit vector $\hat{B}=\vec{B}/|\vec{B}|$ at point $O$?

$\hat{B} = $ $\hat{x} + $ $\hat{y} + $ $\hat{z}$

Problem 3
The black lines in the figure below represent lines of contant electrostatic potential.

How much work is required to move an electron ($m=9.11\times10^{-31}$ kg, $q=-1.6022\times10^{-19}$ C) along the red path from point $A$ to point $B$?

$W=\int\limits_A^B \vec{F}\cdot d\vec{r}$

$W = $ [J]

Problem 4

The position of an airplane travelling at supersonic speed is given by:

Here $t$ is the time in seconds. Where along the $y$-Axis ($x=z=0$) will the sonic boom be heard at $t=0$?

$y=$ [m]

The speed of sound is $c=$ 340 m/s.

Problem 5

Two-dimensional waves are sent out from two point sources at positions $\vec{r}_1=-2\hat{x}$ [m] and $\vec{r}_2=2\hat{x}$ [m]. The amplitude of the waves is $1/\sqrt{|\vec{r}-\vec{r}_j|}$, where $|\vec{r}-\vec{r}_j|$ is the distance from the sources. The wavelength of the waves is $\lambda=$  m and the radial frequency is $\omega=$  rad/s.

What is the speed of the waves?

$v=$ [m/s]

A periodic oscillation is observed at position $\vec{r}=3\hat{x}+3\hat{y}$ [m]. What is the amplitude of this oscillation?

$A=$ [m]

Problem 6

A lens with a focal length of  cm is located at the origin, $(x=0,y=0,z=0)$. The optical axis of the lens is parallel to the $x$-axis.A light emitting diode (LED) is located at $x=-40$ cm, $y=2$ cm, $z=1$ cm. The LED is $\sqrt{5}$ cm from the optical axis. The lens focuses the light to an image point. Where is this image point?

$x =$ cm   $y =$ cm   $z =$ cm