Problem 2: The potential energy for a particular two-dimensional force field is given by V(x, y) = Axe-ky, where the constants A and k are chosen for dimensional consistency. (a) Choose specific (and reasonable) values for A and k, and use Mathematica, Desmos, or the equivalent to plot contours of constant potential energy for various energy levels, over a region of the plane centered on the origin. (b) Show that an infinitesimal displacement along an equipotential line has the form dx dr = dz x + ŷ. kx (c) Find the expression for the force field, and plot this vector field over the same region of the plane (and with the same choice of constants) as in part (a).

Problem 2: The potential energy for a particular two-dimensional force field is given by V(x, y) = Axe-ky, where the constants A and k are chosen for dimensional consistency. (a) Choose specific (and reasonable) values for A and k, and use Mathematica, Desmos, or the equivalent to plot contours of constant potential energy for various energy levels, over a region of the plane centered on the origin. (b) Show that an infinitesimal displacement along an equipotential line has the form dx dr = dz x + ŷ. kx (c) Find the expression for the force field, and plot this vector field over the same region of the plane (and with the same choice of constants) as in part (a).

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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Question

**Problem 2:** The potential energy for a particular two-dimensional force field is given by \( V(x, y) = Axe^{-ky} \), where the constants \( A \) and \( k \) are chosen for dimensional consistency.

(a) Choose specific (and reasonable) values for \( A \) and \( k \), and use Mathematica, Desmos, or the equivalent to plot contours of constant potential energy for various energy levels, over a region of the plane centered on the origin.

(b) Show that an infinitesimal displacement along an equipotential line has the form

\[
dr = dx \, \hat{x} + \frac{dx}{kx} \, \hat{y}.
\]

(c) Find the expression for the force field, and plot this vector field over the same region of the plane (and with the same choice of constants) as in part (a).

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