Introduction to Mathematical Physics/Electromagnetism/Exercises

Exercice:

exoeqhelmoltz

Assume constitutive relations to be:

$D(r,t)=\epsilon (r,t)*E(r,t)$

where $*$ represents temporal convolution\index{convolution} (value of $D(r,t)$ field at time $t$ S depends on values of $E$ at preceeding times) and

$H={\frac {B}{\mu _{0}}}$

Show that in harmonical regime ($E(r,t)={\mathcal {E}}(r)e^{i\omega t}$) and without any charges ${\mathcal {E}}(r)$ field verifies Helmholtz equation :

$\Delta {\mathcal {E}}+k^{2}{\mathcal {E}}=0.$

Give the expression of $k^{2}$.

Exercice:

Show (see ([#References

Exercice:

Proove charge conservation equation eqconsdelacharge from Maxwell equations.

Exercice:

Give the expression of electrical potential created by quadripole $Q_{ij}$.

Exercice:

Show from the expression of magnetic energy that force acting on a point charge $q$ with velocity $v$ is:

$f=qv\wedge B$