Introduction to Mathematical Physics/Electromagnetism/Exercises

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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