Recent content by LeoJakob

For the dipole moment I calculated $$\begin{aligned} M &= \int \rho(\mathbf{r}) \mathbf{r} \times \mathbf{E}(\mathbf{r}) d^{3} \mathbf{r} \\ \mathbf{E}(\mathbf{r}) &\approx \mathbf{E}(\mathbf{0}) + \sum_{i=1}^{3} \nabla E_{i}(\mathbf{0}) \cdot \mathbf{r} \\ \mathbf{M}_{D} &= \mathbf{p} \times...

is there an easier way to calculate the dipole moment? I described ## \vec r## in spherical coordinates. I thought at first that due to the symmetry I can assume that dipole-moment only points in the ##z##-direction, but the charge distribution is inhomogeneous, so I made the following...

Thank you :) $$ \begin{align} \Delta \chi - \frac{1}{c^{2}} \ddot{\chi} &= -\frac{3}{2 c^{2}} \dot{\Phi} = -\frac{3}{2 c^{2}}(-\vec{a} \cdot \vec{r}) = \frac{3}{2 c^{2}}(\vec{a} \cdot \vec{r}) \\ \Leftrightarrow \quad \Box \chi &= -\frac{1}{c^{2}} \frac{3}{2} \dot{\Phi} = \frac{3}{2...

Hello, here is my solution attempt: (i) $$ \begin{aligned} 0 & =\Phi^{\prime}=\Phi-\frac{\partial \chi}{\partial t} \Rightarrow \Phi=\frac{\partial \chi}{\partial t} \\ & \Rightarrow \int \Phi dt=\chi \\ & \Rightarrow \chi=\int \limits_{0}^{t}-(\vec{a} \cdot \vec{r}) t^{\prime} d...

In the headline to the question the statement should have been: rotation of macroscopic magnetization = averege (Magnetization current density ) The Magnetization current densities ##\vec{j}_{\text{mag}}^{(i)}## of individual particles ##i## are stationary ##(\vec{\nabla} \cdot...

$$ \begin{array}{l} m_z = \vec m \cdot \vec{e}_z = \frac{1}{2} \int \limits_{0}^{2 \pi} d \phi \int \limits_{0}^{R} \rho \omega\left(r^{\prime}\right)^{4} d r^{\prime} \int \limits_{0}^{\pi} \sin ^{2} \theta \left[\left(-\vec{e}_{\theta}\right) \cdot \vec{e}_z\right] d \theta=\frac{1}{2} 2 \pi...

$$ \begin{array}{l}m=\frac{1}{2} \int \limits_{V} \vec{r}^{\prime} \times \vec{j}\left(\vec{r}^{\prime}\right) d V \\ \vec{\omega}=\omega \overrightarrow{e_{z}}=\omega\left(\cos \theta \vec{e}_{r}-\sin \theta \overrightarrow{e_{\theta}}\right) \\ \vec{v}=\vec{\omega} \times \vec{r}=\left(\omega...

Ahhhh, ##\vec e_\phi## is zero becaues it is perpendicular to the ##z##- axis., so it will not contribute to a ##z## component. So I'm now at this point: $$ \left(\begin{array}{l}0 \\ 0 \\ 1\end{array}\right)=\lambda\left(\begin{array}{c}\sin \theta \cos \phi \\ \sin \theta \sin \phi \\ \cos...

##(\vec {e}_{r'} \times \vec {e}_{\phi})_z=(-\vec {e}_{\theta})_z=\sin \theta ##? I don't know how I can represent ##\vec e_z## in spherical coordinates?

But ##\vec {e}_{r'} \times \vec {e}_{\phi} = \vec {e}_{\theta}## is correct? True! First I need to calculate ##\vec{j}(\vec{r}, t)=\rho(\vec{r}, t) \vec{v}(\vec{r}, t)## Failed attempt: I tried to use spherical coordinates. Because of the radial symmetrie the magnetic moment will only be...

I'm really sry for that, I meant the heaviside-function and should have wrote ##\vartheta## for the spherical coordinates. I will give you a new calculation in a few minutes

Ich wäre Ihnen sehr dankbar, wenn Sie sich meine Lösung der folgenden Übung ansehen: A sphere with radius ##R ## is spatially homogeneously loaded and rotates with constant angular velocity ##\vec{ \omega}## around the ##z ## axis running through the center of the sphere. Calculate the...

The following is an example from my script. I always have trouble identifying useful symmetries. Can someone explain to me why (for example) the vector potential doesn't have a ##z## dependence? I understand that there is no ##\varphi## dependency. I don't understand why the field of ##\vec{A}##...

Thank you ! :) It is an exercise to solve for the upcoming exams: -segment 1 and 2 produce the same magnetic field at a point on the z axis - a whole circle on the central x-axis would create a magnetic field that points only in the z-direction because of the radial symmetry - a semicircle...

For the following conductor loop, determine the magnetic field along the ##z##-axis, which passes through the center of the conductor loop and is perpendicular to it. The conductor loop consists of an infinitely long wire through which a constant current ##I## runs. Is it possible to determine...