Physics

Physics · electromagnetism

Problem

(a) Knowing that Legendre polynomials are orthogonal (⟨Pm(x),Pn(x)⟩=0\left(\left\langle P_{m}(x), P_{n}(x)\right\rangle=0\right. if m≠n)m \neq n) and deg⁡Pn(x)=n\operatorname{deg} P_{n}(x)=n, obtain P2(x)P_{2}(x) and P3(x)P_{3}(x). For reaching the usual and most convenient form of these polynomials, divide your results by the norm: ∥Pn(x)∥=22n+1\left\|P_{n}(x)\right\|=\frac{2}{2 n+1}. Let us now consider a sphere of radius RR centered at the origin. Suppose a point charge qq is put at the origin and that this is the only charge inside or outside the sphere. Furthermore, the potential is Φ=V0cos⁡θ\Phi=V_{0} \cos \theta on the surface of the sphere. We know that we can write the potential induced by the charge on the sphere (without taking into account qq ) in the following way: Φ−=∑n=0∞AnrnPn(cos⁡θ),r<RΦ+=∑n=0∞Bnrn+1Pn(cos⁡θ),r>R\begin{array}{ll} \Phi_{-}=\sum_{n=0}^{\infty} A_{n} r^{n} P_{n}(\cos \theta), & r<R \\ \Phi_{+}=\sum_{n=0}^{\infty} \frac{B_{n}}{r^{n+1}} P_{n}(\cos \theta), & r>R \end{array} Given constants: Computing electric fields Electrostatics relies on multiple methods for computing electric fields and potentials. In this problem, we will explore two of them, Gauss's Law and Legendre polynomials. Uniform charge distributions Let us consider a hollow conducting sphere of radius RR charged with the electric charge QQ, uniformly distributed on its surface. In order to calculate its potential, we can use Gauss's Law, which states that the flux of the electric field dF=E⋅dAd F=\mathbf{E} \cdot \mathbf{d} \mathbf{A} across a closed surface is proportional to the charge enclosed by that surface: F=Q/ε0F=Q / \varepsilon_{0}. We have denoted dA=dAn\mathbf{d A}=d A \mathbf{n} the elementary oriented (towards the exterior) surface element. Context question: (a) Compute the electric potential inside and outside the sphere. Context answer: Φ−=Q4πε0R\Phi_{-}=\frac{Q}{4 \pi \varepsilon_{0} R},Φ+=Q4πε0r\Phi_{+}=\frac{Q}{4 \pi \varepsilon_{0} r} Extra Supplementary Reading Materials: Legendre polynomials and non-uniform charge distributions Legendre polynomials are a type of orthogonal polynomials essential in mathematical physics. One of their applications is in computing electric potentials for more complicated charge configurations. We will denote the nn-th Legendre polynomial (having degree nn ) as PnP_{n}. Legendre polynomials are defined on [−1,1][-1,1] and we can express their scalar product as ⟨Pm(x),Pn(x)⟩=∫−11Pm(x)Pn(x)dx(1)\left\langle P_{m}(x), P_{n}(x)\right\rangle=\int_{-1}^{1} P_{m}(x) P_{n}(x) d x \tag{1} The first two Legendre polynomials are P0(x)=1P_{0}(x)=1 and P1(x)=xP_{1}(x)=x.

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