Consider the vector field F F(x, y, z) = sin yi + x cos yj + – sin zk. (a) Show this vector field is conservative. (b) For this vector field, find a potential function ø which satisfies o(0, 0, 0) = 2020.
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- The right triangle has base b and height a with uniformly distributed surface charge density σ. The potential at the vertex P is: (answers in the image)A point charge in the amount of Q is placed between an infinite grounded conducting wall to its left at a distance of L and a grounded conducting sphere of radius r to its right at a distance of R (R>r). The line connecting the center of the sphere and the point charge extends leftward to strike the wall at normal incidence. Find the electrostatic potential everywhere to left of the wall but outside the sphere.The positive charge of a dipole with the dipole moment D~ = q ~d is located at the origin of Cartesiancoordinates, O. Consider some point A, such that the vector AO~ = α~d. Show that the electricfield potential, VD, at point A is equal toVD =q4πε0dα(1 + α)
- 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.2.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?
- Consider a charged arc segment with radius R = (3.90x10^-2) m and charge density A = (8.700x10^-6) C/m and a central angle p = (3.4x10^-1) rad. What is the electric potential at the geometrical centrum of the arc segment (point Pin the figure, in the same plane as the arc segment) ? Answer with unit volts (V) using proper scientific notation. •P R(0,0,0) and the 5. The point o is the origin of the coordinate system, o = coordinates of b are b = (x,y, z). The electric potential at o is zero, V, = 0. Hence, the electric potential at b is V = – E · dr. You can take any path from o to b. (a) Here is one particular path from o to b. First move on a straight line from o to a = : (x,0,0), then from a to a' = (x, Y, 0), and finally from a' to b = (x, y, z). Make a plot of this path, indicating the coordinate system and the locations of a, a' and b. – SE • dr depends only on the y However, the line integral V coordinate of b = (x, y, z). Hence, in the following, we will focus on b = (0, y, 0) and take the straight path from o to b. (b) Compute E · dr for y d result, use this potential next.a circular ring having radius R and lying in x-y plane with its center at origin carries a uniformly distributed charge q. calculate the electric potential everywhere for r > R.
- What is the work done by moving in the force field F(x,y) = [6x2 + 2,16y7 ] along the parabola y = x2 from (−1,1)to(1,1) ? In part a) compute it directly. Then, in part b), use the theorem.The charge density on a disk of radius R = 11.2 cm is given by σ = ar, with a = 1.34 μC/m³ and r measured radially outward from the origin (see figure below). What is the electric potential at point A, a distance of 48.0 cm above the disk? Hint: You will need to integrate the nonuniform charge density to find the electric potential. You will find a table of integrals helpful for performing the integration. V R AConsider a sphere of radius R, carrying a charge density p(r) = ar. The total charge of the sphere is a known value Q. (a) Calculate the constant a and the electric field (maanitude and direction) inside and outside the sphere, as a function of Q and R. (b) Calculate the potential in the origin, as a function of Q and R. (c) Suppose now that the charge Q is uniformly distributed on the surface of the same sphere. Calculate the difference of electrostatic energy between the final and initial configurations.