QQuestionAnatomy and Physiology
QuestionAnatomy and Physiology
Fo ; 2 The potential on the Wie of a sphere of radius R is given by the following:
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~~ b) What is the general form of the potential equation?
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Answer
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Step 1:Q^2a: Identify which l's are present
V(R,\theta) = 12V_0 - 6V_0(\cos^2\theta + \cos\theta)
The terms \cos\theta and \cos^2\theta correspond to Legendre polynomials P_1(\cos\theta) and P_2(\cos\theta). The constant term is P_0(\cos\theta). So, l = 0, 1, 2 are present.
Step 2:Q^2b: General form of the potential equation
V(r,\theta) = \sum_{l=0}^{\infty} \left( A_l r^l + \frac{B_l}{r^{l+1}} \right) P_l(\cos\theta)
The general form for the potential in spherical coordinates (no azimuthal dependence) is a sum over Legendre polynomials with coefficients for inside and outside the sphere.
Step 3:Q^2c: Find the A_l coefficients
V(R,\theta) = A_0 P_0(\cos\theta) + A_1 R P_1(\cos\theta) + A_2 R^2 P_2(\cos\theta)
Expand the given potential in terms of Legendre polynomials: P_0 = 1, P_1 = \cos\theta, P_2 = \frac{1}{2}(3\cos^2\theta- 1). Match coefficients to solve for A_0, A_1, A_2.
Step 4:Q^2c: Calculate A_l values
A_0 = 9V_0,\quad A_1 = -6V_0/R,\quad A_2 = 4V_0/R^2
Matching coefficients gives A_0 = 12V_0 - 3V_0 = 9V_0, A_1 = - 6V_0 /R, A_2 = 4V_0 /R^2.
Step 5:Q^2d: Find the B_l coefficients
B_l = 0$$ for all l
Since the potential is specified on the surface and we want a finite solution at the origin (inside), all B_l must be zero for the inside solution.
Step 6:Q^2e: Write the potentials inside and outside
V_{\text{in}}(r,\theta) = 9V_0 + (-6V_0)\frac{r}{R}\cos\theta + 4V_0\left(\frac{r}{R}\right)^2\frac{1}{2}(3\cos^2\theta-1) \newline V_{\text{out}}(r,\theta) = \alpha_0\frac{1}{r^0} + \alpha_1\frac{R}{r^2}\cos\theta + \alpha_2\frac{R^2}{r^3}\frac{1}{2}(3\cos^2\theta-1)
The inside potential uses only A_l terms, and the outside would use B_l terms (to be determined by boundary conditions if needed).
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