Analytical mechanics for students of physics and engineering . last equation may be put in the form of an dm rmdm i/0 T (VII) 180. Intensity of the Field. — The intensity at any point ofa potential field, or a field of force, is defined as the /•/( need by a unit mass when placed at that point. Let H denote the intensity at a point. Then, if F is theforce experienced by a mass m when placed at that point,we have, by definition, W = —,i (VIII) m and // = —, in = --,(-) III \dS ) 8»ta7 FIELDS OF FORCE AND NEWTONIAN POTENTIAL 213 Similarly and = dV ds Hx = dVto Hy = dV ay
Analytical mechanics for students of physics and engineering . last equation may be put in the form of an dm rmdm i/0 T (VII) 180. Intensity of the Field. — The intensity at any point ofa potential field, or a field of force, is defined as the /•/( need by a unit mass when placed at that point. Let H denote the intensity at a point. Then, if F is theforce experienced by a mass m when placed at that point,we have, by definition, W = —,i (VIII) m and // = —, in = --,(-) III \dS ) 8»ta7 FIELDS OF FORCE AND NEWTONIAN POTENTIAL 213 Similarly and = dV ds Hx = dVto Hy = dV ay dz (IX) Therefore the component, along any direction, of the intensityat any point equals the rate at which the potential diminishesat that point as one moves along the given direction. ILLUSTRATIVE EXAMPLES. 1. Find the expressions for poten-tial and intensity at a point due toa spherical shell. Let P, Fig. 107, be the point andR its distance from the center of theshell. Then taking a zone for theelement of mass, as shown in thefigure, we get. Fig. 107. andTherefore dm = a. Therefore theexpression for the potential may be put in the form v=_yrjJIa[{a + R)_iR_a)] T-iira2m Therefore outside the shell the potential is the same as if the mass of theshell wnr concentrated at its center. (b) Po» i Within the Sphere. —In this case R a. = 0 when R < a. Therefore the shell attracts8 particle which is outside
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