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Legendre polynomials

A Dielectric Sphere in a Homogeneous Electric Field
In this problem we will encounter the main physical features of dielectric spheres  their induced field and polarizability. The result is of great interest: understanding the interaction of light with small particles is one of the main concerns of nanophotonics. It leads to astonishing applications like cloaking, improvement of solar cells and lithography with extreme resolution.

A Magnetic Sphere with Surface Current
Using the magnetostatic potential can be extremely useful to calculate magnetostatic problems. However, it can only be defined if no currents are present. In this problem you will discover how we can still use the potential in situations of strongly confined surface currents.

Multipole Moments in Rotational Symmetry
In rotational symmetry, the multipole moments of a charge distribution are described by just one component per order l. This is a drastic difference to the usual 2l + 1 independent components. Get to know this simplifications with two examples and derive it for any charge distribution with rotational symmetry!