# 15_2_11 - 11. The scalar potential for the two-source...

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11. The scalar potential for the two-source system is φ( x , y , z ) = φ( r ) = - m | r - k | - m | r + k | . Hence the velocity field is given by v ( r ) = φ( r ) = m ( r - k ) | r - k | 3 + m ( r + k ) | r + k | 3 = m ( x i + y j + ( z - ‘) k ) [ x 2 + y 2 + ( z - ‘) 2 ] 3 / 2 + m ( x i + y j + ( z + ‘) k ) [ x 2 + y 2 + ( z - ‘) 2 ] 3 / 2 . Observe that v 1 = 0 if and only if x = 0, and v 2 = 0 if and only if y = 0. Also v ( 0 , 0 , z ) = m ± z - | z - | 3 + z + | z + | 3 ² k , which is 0 if and only if z = 0. Thus v = 0 only at the origin. At points in the xy -plane we have v ( x , y , 0 ) = 2 m ( x i + y j ) ( x 2 + y 2 + 2 ) 3 / 2 . The velocity is radially away from the origin in the xy -plane, as is appropriate by symmetry. The speed at ( x , y , 0 ) is v( x , y , 0 ) = 2 m p x 2 +
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