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44 COULOMB'S LAW

Example

I f we express all positions in rectangular coordinates, we can easily write down an

explicit form for (2-10). Using (1-13) and (1-14), and noting that the various charges are

designated by the subscripts / rather than by primes, we find that (2-10) becomes

f qq, [ ( * - * , ) * + (y- y t ) 9 . + { ' - * , ) * ] . .

In a sense, (2-12) provides a simple recipe for solving problems, since once the values of

all the charges and their positions in rectangular coordinates are given, all that remains

is to substitute these numbers into (2-12) and to simplify the result as much as possible.

4 CONTINUOUS DISTRIBUTIONS OF CHARGE

We often encounter situations in which the other charges are so close together

compared to the other distances of interest that we can regard them as being

continuously distributed, much as we can treat a glass of water, on a laboratory scale,

as a continuous distribution of mass by neglecting its molecular structure. We can deal

with such a case by considering a region of the charge distribution that is so small that

the charge within it can be written as dq' and treated as a point charge; this is

illustrated in Figure 2-4. We can still use (2-10), but now the sum wil l become an

integral over the complete charge distribution so that

a , dq'R

F < = i / " k < 2 - 1 3 >

where (2-2) continues to be applicable.

I f the charges are distributed throughout a volume, we can introduce a v o l u m e

charge density p, which is defined as the charge per unit volume and hence wil l be

measured in coulombs/(meter) 3. (We wil l write this charge density as p c h in the

infrequent cases in which it might be confused with the p of cylindrical coordinates.)

Then the charge contained in a small source volume dr' wi l l be given by

dq' = p ( r ' ) dr' (2-14)

dq-

Figure 2-4. Charge element of a continu-

ous distribution.

44 COULOMB'S LAW

Example

I f we express all positions in rectangular coordinates, we can easily write down an

explicit form for (2-10). Using (1-13) and (1-14), and noting that the various charges are

designated by the subscripts / rather than by primes, we find that (2-10) becomes

f qq, [ ( * - * , ) * + (y- y t ) 9 . + { ' - * , ) * ] . .

In a sense, (2-12) provides a simple recipe for solving problems, since once the values of

all the charges and their positions in rectangular coordinates are given, all that remains

is to substitute these numbers into (2-12) and to simplify the result as much as possible.

4 CONTINUOUS DISTRIBUTIONS OF CHARGE

We often encounter situations in which the other charges are so close together

compared to the other distances of interest that we can regard them as being

continuously distributed, much as we can treat a glass of water, on a laboratory scale,

as a continuous distribution of mass by neglecting its molecular structure. We can deal

with such a case by considering a region of the charge distribution that is so small that

the charge within it can be written as dq' and treated as a point charge; this is

illustrated in Figure 2-4. We can still use (2-10), but now the sum wil l become an

integral over the complete charge distribution so that

a , dq'R

F < = i / " k < 2 - 1 3 >

where (2-2) continues to be applicable.

I f the charges are distributed throughout a volume, we can introduce a v o l u m e

charge density p, which is defined as the charge per unit volume and hence wil l be

measured in coulombs/(meter) 3. (We wil l write this charge density as p c h in the

infrequent cases in which it might be confused with the p of cylindrical coordinates.)

Then the charge contained in a small source volume dr' wi l l be given by

dq' = p ( r ' ) dr' (2-14)

dq-

Figure 2-4. Charge element of a continu-

ous distribution.