Part 3: Other Types of Densities

In general, a density function is a function which relates physical quantities to geometric quantities by specifying units of a certain physical quantity per unit volume. For example, in electrostatics we often consider charge densities, where a charge density r( x,y,z) is the amount of charge per unit volume near a point ( x,y,z) in space. It follows that the amount of charge dQ in a small region of space with volume dV is given by

dQ = r( x,y,z) dV

and as a result, if W is a region in space which completely contains a ``charge cloud,'' then the total charge Q in W is given by

Q =  dQ =  r( x,y,z) dV

EXAMPLE 5    Compute the total charge in a sphere centered at the origin with radius R = 2 if the charge density is given by

r( x,y,z) = 2z Coulombs per cubic meter

Solution: Since the equation of the sphere is x²+y²+z² = 4, solving for z leads to

z = ±   4-x2-y2

As a result, the total charge is

Q =   2zdV =  ó õ Ö 4-x2-y2 -Ö 4-x2-y2   2zdz  dA

where R is the circle in the xy-plane with radius 2 centered at the origin. Thus,

Q =   z2  Ö 4-x2-y2 -Ö 4-x2-y2 dA = 0

That is, the equal number and symmetric distribution of positive and negative charges results in a net total charge inside the sphere of 0.

       

Other densities an be derived from mass and charge densities. For example, the potential energy U due to of the force of gravitational attraction between two point masses with mass M and m, respectively, is given by

U = -G  Mm r

where r is the distance between the two points and G is the universal gravitational constant. Thus, a small section of a solid S with mass dM has a potential energy of

dU = -G  m r   dM

on an object with mass m which is at a distance r from the small section.

Thus, if the solid has a mass density m( x,y,z) , then the potential energy of a small section containing the point (x,y,z) is approximately

dU = -G  m r m( x,y,z) dV

and the total gravitational potential energy of the solid is

U =   dU = -Gm  mdV r

Finally, if the mass m is located at the point ( a,b,c) , then the distance between the point masses is

r =   ( x-a) 2+( y-b) 2+( z-c) 2

Thus, the total gravitational potential due to a solid S with a mass-density of m( x,y,z) is given by

U = -Gm  m( x,y,z) dV ( x-a) 2+( y-b) 2+( z-c) 2

EX AMPLE 6   What is the potential energy of a mass m located at the point ( 0,0,r) due to the gravitational attraction of a sphere of radius R centered at the origin with a constant mass density.

Solution: Assuming m is constant and substituting the location ( 0,0,r) leads to

U = -Gm  mdV ( x-a) 2+( y-b) 2+( z-c) 2

where S is the sphere with equation x²+y²+z² = R².