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Unformatted text preview: 1. Is it correct to say that ~ ~ A is a vector field that is perpendicular to ~ A because it is a cross product of ~ and ~ A ? Answer: Answer is NO! For example, one can show ~ ~ A is not necessarily perpendicular to the vector ~ A , as one would expect from a cross product. We will show this by an explicit construction. Choose ~ A =- ay i + bx j + c k ; where a , b and c are non-zero positive constants. ~ ~ A = ( a + b ) k . The scalar product of ~ ~ A and ~ A is ( a + b ) c 6 = 0. The field lines of the vector field ~ A look like a cork screw and its curl points towards the advancing direction of the screw given by the right hand rule. On the other hand the field also has a non-zero component in the direction in which the screw advances. It is not surprising that they are not perpendicular. What then makes ~ ~ A a vector field? ~ ~ A is a vector field because its components transform like one. For a similar reason ~ ~ A is not a simple dot product. For example, ~ ~ A 6 = ~ A ~ . It is a scalar field because it does not change under a general rotation in the three dimensional space. 1 2. A charge Q is distributed uniformly over a sphere of radius R. Express the resulting volume charge density involving Dirac- function. Answer: The charge density is independent of the angles and and a function of r alone, i.e. ( ~ r ) ( r ). Charge density is zero for all values of r except at r = R . When charge density is integrated over all space (integration over the entire domains of and gives 4 ,) one must get the total charge Q . Thus the integration of ( r ) over r is finite even though the function is nonzero only ar r = R . It is possible only if charge density has an integrable....
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- Spring '09