Consider the vector field F(x,y)=(4x^3y^2−2xy^3)i+(2x^4y−3x^2y^2+4y^3) a. Find the potential function f(x,y). b. Determine the work done by the vector field in moving a particle along a curve C from (1,2) to (3,1).

3. Given the following scalar potentials (V), calculate the solution for the gradient of V (VV), and plot the vector arrow representation of this vector field over the given limits. (a) V = 15 + r cos o, for 0 < r < 10, and 0 < $ < 2n. (b) V = 100 + xy, for –10 < x < 10,

Please explain  An electrically charged particle is held fixed at the origin. Let V = 1/r be the potential energy of a second particle with unit electric charge located at (x, y, z), where r = sqrt(x^2 + y^2 + z^2). The force on this second particle is given by −∇V . Calculate the force on the particle when it is at (1, 2, −2). Show working

Homework Q.1

Consider a particle of mass m acted upon by a central potential (r is distance and a>0 is a constant): U (r) p2 a) What are the constants of the motion? b) Is a circular orbit possible in this potential? If so, state the conditions for it. c) Under what conditions will the particle fall to the center of the potential (r = 0)?

Consider a parallel plate capacitor as shown in the figure. A proton is placed near the positive plate and let go from rest. Sketch the electric field lines of the parallel plate capacitor and the variation of the following parameters as a function of position x inside of the capacitor: Electric potential produced by the capacitor (V), Potential Energy of the proton(U), Total Energy of the proton (Etot), and Kinetic Energy of the proton (K). -Q V U Ett Figure 1

d and e

Consider the 3-dimensional force field ⃗ F = (x^2 − ze^y)⃗i + (y^3 − xze^y)⃗j + (z^4 − xe^y)⃗k:(a) Show that ⃗ F is conservative.(b) If ⃗ F is conservative, find the corresponding potential function f (x, y).(c) If an object travels on a path ⃗r (t), (t_0 < t < t_f ), does the work done bythe force field depend on the path taken? Find the work done on the object movingon the path ⃗r (t) by the force field ⃗ F , if ⃗r (t_0) = (3, 1, 2) and ⃗r (t_f ) = (6, 2, 5)