Two force fields act in the plane

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Vector calculus and field geometry · Vector calculus and field geometry

Mathematical Methods for Physics, Ch Line, surface and volume integrals

Problem

Two force fields act in the plane z=0z=0, with xx and yy in metres: FA=(2y, x, 0) N,FB=(y, x, 0) N.\mathbf{F}_A = (2y,\ x,\ 0)\ \mathrm{N}, \qquad \mathbf{F}_B = (y,\ x,\ 0)\ \mathrm{N}. A small object is moved slowly from O=(0,0,0) mO=(0,0,0)\ \mathrm{m} to P=(2,3,0) mP=(2,3,0)\ \mathrm{m}. ((a)) For each field, find the work it does along route 1, O→(2,0,0)→PO\to(2,0,0)\to P, and along route 2, O→(0,3,0)→PO\to(0,3,0)\to P, each made of two straight legs. ((b)) Compute the curl of each field, and say which field's work between two points is the same on every route. ((c)) For that field, find a potential φ\varphi with F=−∇φ\mathbf{F}=-\nabla\varphi, and use it to give the work along the straight line from OO to PP without parameterising that line.

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