A sphere of radius R, centered at the origin, carries charge density ρ(r, θ) = kRr^2(R − 2r ) cos θ where k is a constant, and r, θ are the usual spherical coordinates. Find the approximate potential for points on the z axis, far from the sphere.
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A sphere of radius R, centered at the origin, carries charge density
ρ(r, θ) = kRr^2(R − 2r ) cos θ
where k is a constant, and r, θ are the usual spherical coordinates. Find the
approximate potential for points on the z axis, far from the sphere.
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- A thin-shelled hollow sphere of radius R has a uniform surface charge density σ. Fixed at its center is a point charge q. a) Use Gauss's law to find the electric field a distance r from the center. (Give answers for r < R and r > R.) b) Taking the electric potential V to vanish at infinity, find the electric potential as a function of r, the distance from the center. (Give answers for r < R and r > R.)Please answer (a), (b), and (c).Compute the potential at B(1, 2, 5) if potential at C(0, -1, -1) is 8 kilovolts in a region in vacuum where the electric field is due to a uniform infinite line charge of density -20 microcoulombs per meter located at y = -4, z = 3. All coordinates are measured in meters.
- A dielectric sphere in an external field. Consider a simple dielec- tric with permittivity e, in the form of a uniform spherical ball of radius a. It is placed at the origin in an external electrostatic potential (x, y, z) = bxy (where r, y, z are Cartesian coordinates and b is a constant). Find the elec- trostatic potential o and electric field E everywhere. %3DConsider a circular arc of constant linear charge density A as shown below. What is the potential V, at the origin O due to this arc?Given an electric potential of find its corresponding electric field vector. Sol. Using the concept of potential gradient, we have Since we only have, radial direction (r). Then, the del-operator will be By substitution, we get Evaluating the differential, we, get the following *(q/)
- Solve ASAPConsider two spherical infinitely thin conducting surfaces (= ∞o) with the same center. The cross section of this configuration is shown below. The inner sphere (radius a) has a total negative charge of -Q. The outer sphere (radius b) has a positive charge with unknown amount. Assume the electric potential at the center is Vo (Vo > 0) and at infinity is zero. Also, assume free-space permittivity in all regions. (a) Find the electric potential V(R) at the distance R from the center for 0 < R < b. (b) Find the amount of positive charge on the outer sphere. Hint: The integral form of Gauss' Law is your best friend! Also you may pay close attention to the net charge.Compute for the absolute potential for every of the following distances from a charge of 2 μC: whenr = 100 mm and r = 500 mm. How much work is needed to carry a 0.05 μC charge from the point at r =100 mm to that at r = 100 mm?
- Consider a thin, uniformly charged rod of length L with total charge Q and test points A, a distance a from the center of the rod and B a distance b from the rod. В Find the potential difference between A and B first by integrating the point source potential to find V and V and subtracting, and then by integrating the field. Compare the results in the limit of L>>(a and b). To test the far field limit, compare the appropriate result to the case where L is much less than both a and b. You may need to do this one numerically.The 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 1/²What is the potential at the center of a disk with a hole cut out of its center with an inside radius a and outside radius b and a surface charge density of d?