# Introduction to Mathematical Physics/Electromagnetism/Exercises

Exercice:

exoeqhelmoltz

Assume constitutive relations to be:

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

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

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

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

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

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

Exercice:

Show (see ([#References

Exercice:

Proove charge conservation equation eqconsdelacharge from Maxwell equations.

Exercice:

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

Exercice:

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

${\displaystyle f=qv\wedge B}$