I have a question about the solution to this problem ''find the flux of x hat(i)+y hat(j)+z hat(k) through the sphere of radius a and center at the origin. Take n pointing outward.''

cimithe4c 2022-10-19 Answered
I have a question about the solution to this problem ''find the flux of x i ^ + y j ^ + z k ^ through the sphere of radius a and center at the origin. Take n pointing outward.''
The answer in the book was, we have n = ( x i ^ + y j ^ + z k ^ ) a ; therefore F . n = a and then they integrate it, but what I don't get is how F . n = a isn't the vector n the same vector as F but scaled by 1 / a so the dot product must be ( x 2 i ^ + y 2 j ^ + z 2 k ^ ) a
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Answers (2)

silenthunter440
Answered 2022-10-20 Author has 19 answers
A dot product is a scalar but the expression you guessed is a vector.
Ponder
( x i ^ + y j ^ + z k ^ ) ( x i ^ + y j ^ + z k ^ ) = x 2 i ^ i ^ + x y i ^ j ^ + x z i ^ k ^ + y x j ^ i ^ + = x 2 + y 2 + z 2 .
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Nigro6f
Answered 2022-10-21 Author has 2 answers
The dot product F n gives you a scalar which represent the length of the projection of F on the normal outer vector n. This means that by definition
F n = F 1 n 1 + F 2 n 2 + F 3 n 3 = x 2 + y 2 + z 2 a ,
and since you are on a sphere x 2 + y 2 + z 2 = a 2 , obtaining F n = a, as wished.
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