Chapter 22a

# Chapter 22a - Scalar and Vector Fields Last semester we...

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Scalar and Vector Fields Last semester we spoke, in passing, of gravity as being a field . We spoke for example of the gravitational field of the earth, without giving this a concrete definition. A scalar field defines a scalar quantity (something described by a single number ) at all points in a region of space. We begin with a definition of scalar and vector fields: A vector field defines a vector quantity (something described by a magnitude and direction ) at all points in a region of space. To understand electricity and magnetism we must correct this, defining fields and describing their use.

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Some examples will make the meaning and distinction clear. Temperature is a scalar quantity defined at any point by a single number . We could measure the temperatures at many points throughout a region and then, interpolating between the points, give a reasonably accurate description of the thermal field . 27 28 27 27 27 28 29 28 27 27 28 27 27 27 28 29 28 27 27 28 27 27 27 28 29 28 27 21 22 21 21 21 20 20 19 19 21 22 21 21 21 20 20 20 21 22 21 20 21 21 20 19 20 21 21 20 20 21 20 19 19 20 20 27 28 27 27 27 28 29 28 27 24 23 22 22 22 22 23 24 24 23 24 25 25 24 23 23 23 25 22 23 24 24 25 23 23 24 24 23 21 22 22 23 24 25 25 25 The temperatures in a region of space constitute a scalar field .
We could do the same for say the barometric pressure, describing the value of the barometric pressure throughout a region of space, to get a barometric pressure field . Anything that we can think to measure, described by a single number , and associated with each point of space , can make up a scalar field . Note that the values at each point can be changing in time so the specific values may be only a snapshot in time. At one instant in time for example, the wind speed at each point in the region of interest would constitute a scalar field . The wind velocity for each point, which includes both the wind speed and its direction would however be a vector field . We can imagine the space filled at each point with a value that gives the wind speed along with an arrow that gives its direction at that point, or perhaps, arrows at all points who’s lengths represent the velocity (i.e. vectors).

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To define a gravitational field consider a spherical body B of mass M. The field will specify the vector force exerted by the body on other masses in the space around it. A spherical body B exerts a force on another body of mass m, as if all of its mass is concentrated at its center. Hence, independent of where m is with respect to B the direction of the vector forces comprising the gravitational field will always point toward the center of B. M m m m m
The magnitude of the force on m will vary with its distance from B to satisfy the inverse square law, 2 mM FG r = But this depends on the mass m not just the body B. What is wanted is something that generally describes the field of B alone . To generalize to a force field we divide both sides by the mass m, defining the magnitude of the gravitational force field at all points of space about B as, G 2 FM mr == A convenient way to think about this is to consider a fictitious point particle, of unit mass (i.e. 1 kg), as a

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Chapter 22a - Scalar and Vector Fields Last semester we...

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