| u | = i A → B → = ( 2 | u | μ cos ⁡ θ 4 π | R → | 3 ) r ^ + ( | u | μ sin ⁡ θ 4 π | R → | 3 ) θ ^ + 0 ϕ ^ .Gauss's Law for Magnetism stats that ▽ ⋅ B → = 0. However, this: ( r ^ ∂ ∂ r + θ ^ r ∂ ∂ θ + ϕ ^ r sin ⁡ θ ∂ ∂ ϕ ) ⋅ [ ( 2 | u | μ cos ⁡ θ 4 π | R → | 3 ) r ^ + ( | u | μ sin ⁡ θ 4 π | R → | 3 ) θ ^ + 0 ϕ ^ ] ≠ 0 .All terms should cancel, but they do not. Where am I going wrong?What I got is: ▽ ⋅ B → = − 6 | u | μ ⋅ r ^ 4 π | R | 4 + | u | μ ⋅ r ^ 4 π | R | 4 + 0 ∴≠ 0 .

Question

|    u      |    =  i            A      →                  B      →        =      (    2          |        u          |                      μ        cos        ⁡        θ                    4        π                  |                                      R            →                                                |                    3                      )              r      ^        +      (          |        u          |                      μ        sin        ⁡        θ                    4        π                  |                                      R            →                                                |                    3                      )              θ      ^        +  0            ϕ      ^          .Gauss&#039;s Law for Magnetism stats that   ▽  ⋅            B      →        =  0. However, this:      (                  r        ^                    ∂              ∂        r              +                            θ          ^                    r              ∂              ∂        θ              +                            ϕ          ^                            r        sin        ⁡        θ                    ∂              ∂        ϕ              )    ⋅      [          (      2              |            u              |                              μ          cos          ⁡          θ                          4          π                      |                                              R              →                                                          |                        3                              )                      r        ^              +          (              |            u              |                              μ          sin          ⁡          θ                          4          π                      |                                              R              →                                                          |                        3                              )                      θ        ^              +    0                  ϕ        ^              ]    ≠  0    .All terms should cancel, but they do not. Where am I going wrong?What I got is:  ▽  ⋅            B      →        =  −            6              |            u              |            μ      ⋅                        r          ^                            4      π              |            R                        |                4              +                    |            u              |            μ      ⋅                        r          ^                            4      π              |            R                        |                4              +  0  ∴≠  0    .

cegielnikmzjkf · Accepted Answer

This is the magnetic field generated by a magnetic dipole. The problem is in the way you calculate   ∇  ⋅      B  . The correct expression is  ∇  ⋅      B    =            1              r        2                                ∂                  (                      r            2                                B            r                    )                            ∂        r              +            1              r        sin        ⁡        θ                        ∂              ∂        θ                  (          B      θ        sin    ⁡    θ    )    +            1              r        sin        ⁡        θ                                ∂                  B          φ                            ∂        φ              .Then  ∇  ⋅      B    =  −  2      |    u      |              1              R        4                        μ      c      o      s      θ              4      π        +  2      |    u      |              1              R        4                        μ      c      o      s      θ              4      π        =  0

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