Consider an ideal stationary magnetic dipole m in a static electric field E. Show that the fields carry momentum p = −ε0μ0(m × E). (8.45) [Hint: There are several ways to do this. The simplest method is to start with p = ε0 ∫ (E × B) dτ, write E = −∇V , and use integration by parts to show

Question

Consider an ideal stationary magnetic dipole m in a static electric field E. Show that the fields carry momentum

[latex]p =-\epsilon_{0} \mu_{0}( m imes E )[/latex] (8.45)

[Hint: There are several ways to do this. The simplest method is to start with [latex]p =\epsilon_{0} \int( E imes B ) d au, ext { write } E =- abla V [/latex] , and use integration by parts to show that

[latex]p =\epsilon_{0} \mu_{0} \int V J d au[/latex] .

So far, this is valid for any localized static configuration. For a current confined to an infinitesimal neighborhood of the origin we can approximate V (r) ≈ V (0) - E(0) · r. Treat the dipole as a current loop, and use Eqs. 5.82 and 1.108.[latex] ]^{21}[/latex]

[latex]\oint d l ^{\prime}= 0[/latex] (5.82)

[latex]\oint( c \cdot r ) d l = a imes c[/latex] (1.108)

Accepted Answer

[latex]p =\epsilon_{0} \int_{ u }( E imes B ) d au=-\epsilon_{0} \int_{ u }( abla V) imes B d au=-\epsilon_{0} \int_{ u }[ abla imes(V B )-V abla imes B ] d au[/latex]

[latex]=\epsilon_{0} \oint_{ S } V B imes d a +\epsilon_{0} \mu_{0} \int_{ u } V J d au=\frac{1}{c^{2}} \int_{ u } V J d au[/latex].

(I used Problem 1.61(b) in the penultimate step. Here ν is all of space, and S is its surface at infinity, where B = 0, so the surface integral vanishes.) Using

V( r ) \approx V( 0 )+( \nabla V) \cdot r =V( 0 )- E ( 0 ) \cdot r

,

[latex]p =\frac{1}{c^{2}} V( 0 ) \int J d au-\frac{1}{c^{2}} \int[ E ( 0 ) \cdot r ] J d au[/latex] .

For a current loop, [latex]\int J d au ightarrow \int I d l=I \int d l = 0 [/latex] , and (Eq. 1.108):

[latex]\int[ E ( 0 ) \cdot r ] J d au ightarrow \int[ E ( 0 ) \cdot r ] I d l=I \int[ E ( 0 ) \cdot r ] d l =I a imes E ( 0 )= m imes E[/latex] .

So

[latex]p =-\frac{1}{c^{2}}( m imes E )[/latex] .

Question 8.20: Consider an ideal stationary magnetic dipole m in a static e...

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