A sphere of radius R centered at the origin carries charge density ρ(r, θ) = k*(R/r^ 2) (R − 2r) sin θ where k is a constant, and r, θ are the usual spherical coordinates. Use the multipole expansion to find the approximate potential for points on the z-axis far from the sphere (z >> R).
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A sphere of radius R centered at the origin carries charge density ρ(r, θ) = k*(R/r^ 2) (R − 2r) sin θ where k is a constant, and r, θ are the usual spherical coordinates. Use the multipole expansion to find the approximate potential for points on the z-axis far from the sphere (z >> R).
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- (a) Find x L limit of the potential of a finite uniformly charged rod and show that it coincides with that of a point charge formula, (b) Why would you expect this result?The general solution for the potential (spherical coordinates with azimuthal symmetry) is: +∞ ΣAir² + P₁(Cos 0) B₁ pl+1 l=0 V(r, 0) = Consider a specific charge density (0) = k cos³0, where k is constant, that is glued over the surface of a spherical shell of radius R. Solve for the potential inside the sphere. Hint: Express the surface charge density as a linear combination of the Legendre polynomials.A thin uniformly charged arc with radius R= 2 cm is shown in the figure below. The total charge of the arc is Q= 81 µC. Point 0 is the origin at the center of the arc. R Calculate the linear charge density 1. Write down an (a) expression for the infinitesimal charge dq. Write down the potential dV at point O due to the infinitesimal charge dq.
- Show that electric potential for a shell whose radius is R, has charge q uniformly distributed on its entire surface, is the same as electric potential for a conductor (Solid) has radius R and charge q.Find the potential on the z axis (0,0,h) produced by an annular(s) ring of uniform surface charge density p, in free space. The ring occupies the region z= 0, p≤a, 0≤ ≤27 in cylindrical coordinates.Two large charged plates of charge density ±17μC/m² face each other at a separation of 7 mm. Choose coordinate axes so that both plates are parallel to the xy plane, with the negatively charged plate located at z = 0 and the positively charged plate at z = +7 mm. Define potential so that potential at z = 0 is zero (V (z = 0) = 0). Hint a. Find the electric potential at following values of z: o potential at z = -7 mm: V(z = -7 mm) o potential at z = +2 mm: V(z = +2 mm) o potential at z = +7 mm: V(z = +7 mm) - o potential at z = +16.8 mm: V(z = +16.8 mm) = V. V. V. V. b. An electron is released from rest at the negative plate. With what speed will it strike the positive plate? The electron will strike the positive plate with speed of m/s. (Use "E" notation to enter your answer in scientific notation. For example, to enter 3.14 × 10¹², enter "3.14E12".)
- A disk of radius R has a nonuniform surface charge density sigma = Cr, where C is a constant and r is measured from the center of the disk. Find (by direct integration) the potential at P.A ring of charge of radius a lies in the z = 0 plane and centered on the z-axis. The charge density on the ring is given by p(') = Peo cosp' [C/m]. First, find the electric field at any point on the z-axis, Ē(z). Next, find the potential Þ(z) on the z-axis. Explain why the field in this problem cannot be found by taking the gradient of your answer for Þ(z). xThe potential of a thin spherical shell of radius R is given as V(R, 0) = 3 cos² 0 + cos 0 - 1. Both inside and outside the sphere, there is empty space with no charge density. The questions on this page are based on this system. What is the linear combination of the two Legendre polynomials that will generate this potential (Denote a Legendre Polynomial as P₁, where I is the index of the polynomial starting from 1 = 0)? O a. 2P₁ + P₂ O b. 2P3 + P₁ O c. Po + P₁ O d. None of the listed answers. O e. 2P₂ + P1₁ Of. P3 + P2 What is the radial part of the potential V(r, 0) inside the spherical shell for the Pi with the lowest l (i.e. the pre-factor of Pi)? O a. r O b. None of the listed options. O c. r/R O d. 2 O e. O f. (r/R)² 1/²Using integration, calculate the electric potential at any point along the z-axis. Show all relevant physics and integration details. Hint: Choose an appropriate charge element dQ on the cylinder and find the dV due to that on the z-axis. Then integrate this potential from -L/2 to L/2 along z.What is the electric potential created by an arc with an irregular linear charge density λ = λ*Cos(Q), distributed with a radius of r = R at point O?Using the general expression V(r,0) E (A,r BỊ + 1) P(cose) for the potential at a distance r from the center of a sphere shell of radius R. The acceptable expressions for potentials inside and outside the shell are: V (r,0) : E, Ar' P,(cose) (rSEE MORE QUESTIONSRecommended textbooks for youPrinciples of Physics: A Calculus-Based TextPhysicsISBN:9781133104261Author:Raymond A. Serway, John W. JewettPublisher:Cengage LearningPrinciples of Physics: A Calculus-Based TextPhysicsISBN:9781133104261Author:Raymond A. Serway, John W. JewettPublisher:Cengage Learning