Calculating potential using method of separation of variables.
A sphere of radius R has a specified potential at it’s surface that is given by:
V (R, θ) = kR /epsilon0 (3 cos^2 θ − 1) .
a) Using the method of separation of variables in spherical coordinate, solve Laplace’s equation to find the potential inside and outside.of the sphere. Refer to Griffith’s examples 3.6 and 3.7 for the method and on how to ”eye-ball” the coefficients in the general solution. (10 points)
Using the continuity equation, find the surface charge density σ(θ) that creates the potential you have calculated. (5 points)
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A conducting sphere of radius a, at potential Vo, is surrounded by a thin concentric spherical...
A conducting sphere of radius a, at potential Vo, is surrounded by a thin concentric spherical shell of radius b, over which someone has glued a surface charge density σ(8)-k cos θ, where k is a constant and θ is the polar spherical coordinate. (a) Find the potential in each region: (i)r > b, and () a<r<b. [5 points] [Hint: start from the general solution of Laplace's equation in spherical coordinates, but allow for different coefficients in the radial part...
1. The potential at the surface of a sphere is kept at potential V(R.0)-Vo sin20. The...
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Change of Variables When working integrals, it is wise to choose a coordinate system that fits the problem; e.g. polar coordinates are a good choice for integrating over disks. Once we choose a coordinate system we must figure out the area form (dA) for that system. For example, when switching from rectangular to polar coordinates we must change the form of the area element from drdy to rdrd0. To determine that rdrde is the correct formula how the edges of...
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What is the difference in between conductor and insulators? Write with necessany figures and examples of...
What is the difference in between conductor and insulators? Write with necessany figures and examples of electrostatic charging by Induction? What is conservation of Charge? 1. Problem-1: Find the charge (Q) of a system having 1000 electrons? Explain the electric field produced due to a positive and negative point charges separately with necessary figures? 2. Problem-2: Calculate the electricfield (E) at a field point of 0.2 μm from a point charge q 10 pC? 3. What is electric dipole moment?...
Consider a cylindrical capacitor like that shown in Fig. 24.6. Let d = rb − ra...
Consider a cylindrical capacitor like that shown in Fig. 24.6. Let d = rb − ra be the spacing between the inner and outer conductors. (a) Let the radii of the two conductors be only slightly different, so that d << ra. Show that the result derived in Example 24.4 (Section 24.1) for the capacitance of a cylindrical capacitor then reduces to Eq. (24.2), the equation for the capacitance of a parallel-plate capacitor, with A being the surface area of...
A conducting sphere of radius a, at potential Vo, is surrounded by a thin concentric spherical shell of radius b, over which someone has glued a surface charge density σ(8)-k cos θ, where k is a constant and θ is the polar spherical coordinate. (a) Find the potential in each region: (i)r > b, and () a<r<b. [5 points] [Hint: start from the general solution of Laplace's equation in spherical coordinates, but allow for different coefficients in the radial part...
1. The potential at the surface of a sphere is kept at potential V(R.0)-Vo sin20. The potential at infinity is zero. (a) Find V(r, 0) inside the sphere. (b) Find V(r,0) outside the sphere. (c) Find σ(θ), the charge density on the sphere. (d) Find the total charge of the sphere. (e) The problern would be a lot harder if the potential were specified to be V(R,θ)-Võsin θ Why? Explain how you would do part (a) without going through the...
2. Potential Inside a Sphere We are interested in the electric potential inside a spherical shell that is radius a and centered on the origin. There are no charges inside the she, so the potential satisfies the Laplace equation, However, there is an external voltage applied to the surface of the shell which holds the potential on the surface to a value which depends on θ: As a result, the potential Ф(r,0) -by symmetry, it does not depend on ф-is...
Here's an ODE that will emerge early in Physics 330. Using separation of variables (the PDE 3. version) we can find the following equation for the radial part R(r) of the electric potential for a spherically symmetric charge distribution: r2drR + 2r-R-1(1 + 1)R = O A. Test a solution of the form: R(r)-Arı + Br-(1+1) and verify that it is a solution. B. The constants A and B are determined using boundary conditions i. Imagine that the region of...
Change of Variables When working integrals, it is wise to choose a coordinate system that fits the problem; e.g. polar coordinates are a good choice for integrating over disks. Once we choose a coordinate system we must figure out the area form (dA) for that system. For example, when switching from rectangular to polar coordinates we must change the form of the area element from drdy to rdrd0. To determine that rdrde is the correct formula how the edges of...
What is the difference in between conductor and insulators? Write with necessany figures and examples of electrostatic charging by Induction? What is conservation of Charge? 1. Problem-1: Find the charge (Q) of a system having 1000 electrons? Explain the electric field produced due to a positive and negative point charges separately with necessary figures? 2. Problem-2: Calculate the electricfield (E) at a field point of 0.2 μm from a point charge q 10 pC? 3. What is electric dipole moment?...
What is the difference in between conductor and insulators? Write with necessany figures and examples of electrostatic charging by Induction? What is conservation of Charge? 1. Problem-1: Find the charge (Q) of a system having 1000 electrons? Explain the electric field produced due to a positive and negative point charges separately with necessary figures? 2. Problem-2: Calculate the electricfield (E) at a field point of 0.2 μm from a point charge q 10 pC? 3. What is electric dipole moment?...
Consider a cylindrical capacitor like that shown in Fig. 24.6. Let d = rb − ra be the spacing between the inner and outer conductors. (a) Let the radii of the two conductors be only slightly different, so that d << ra. Show that the result derived in Example 24.4 (Section 24.1) for the capacitance of a cylindrical capacitor then reduces to Eq. (24.2), the equation for the capacitance of a parallel-plate capacitor, with A being the surface area of...
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