5. Consider the vector field given by Ä× R F = (a) Verify that F is a solenoidal vector field. (b) Find a vector potential G.
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(a) Verify that F is a solenoidal vector field. (b) Find a vector potential G .


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- A sphere has radius R and uniform charge density ρ. Take your reference point at infinity and find the potential at all points in space, inside and outside the sphere, then draw your results.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.Consider 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.
- An uncharged, infinitely long conducting cylinder of radius a is placed in an initially uniform clectric ficld E = Eoi, such that the cylinder's axis lics along the z axis. The resulting clectrostatic potential is V (x, y, z) = Vo for points inside the cylinder, and Ega?x x² + y? V (r, y, z) = Vo – Egx + for points outside the cylinder, where Vo is the (constant) clectrostatic potential on the conductor. Usc this expression to determine the resulting clectric ficld, E.Consider an “anti-gravity” potential (or, more mundanely, the electrostatic potential between two like charges), V(r) = α / r where α > 0. Show that circular, elliptical and parabolic orbits do not exist.Please answer (a), (b), and (c), showing all work.
- 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. Find the potential difference between A and B first by integrating the point source potential to find VA and VB 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.Only 2 c)Consider a hollow sphere with radius R and surface potential V(0) = 5 – cos0. Use the general solution of Laplace equation V (r, 8) = E3 (A;r' +)Pi(cos®) to find an expression for the potential at any point outside the sphere. a) V(r,8) : R3 cose R3 cose 2R cose b) V(r, 8) 3R c) V(r,6) R3 R cose 4R 5R d) V (r, 8) - cose e) V(r,8) a b C d e
- 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?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.