On the magnetic dipole radiation in Griffith's book

In summary, the conversation discusses the appearance of magnetic dipole radiation in section 11.1.3 of Griffith's "Introduction to Electrodynamics 4Ed" and the resultant equation which shows that there is no magnetic field in the axis of a wire loop when theta=0. However, the person is missing the fact that although the magnetic flux density is at maximum value there, it is time-varying due to the alternating current. This is because Griffiths is considering a radiating, harmonically oscillating current loop, leading to slower decreasing "radiation terms" in the fields. Working problem 11.5 on page 477 may help clarify this concept.
  • #1
a1titude
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Homework Statement
It's not a homework. I just saw the resultant equation to find that it's strange.
Relevant Equations
$$\mathbf{B} = \mathbf{\nabla} \times \mathbf{A} = - \frac {\mu_0 m_0 \omega^2} {4 \pi c^2} \left( \frac {\sin \theta} {r} \right) \cos [\omega (t - r/c)] \hat{\mathbf{\theta}}$$
In 11.1.3 of Griffith's "Introduction to Electrodynamics 4Ed" appears magnetic dipole radiation, which results in the equation above. According to the resultant equation, there is no magnetic field in the axis of the wire loop because theta=0. However, I think the magnetic flux density is at maximum value there although its time-varying due to the alternating current. What am I missing now? Thanks for your concerns in advance.
 

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a1titude said:
In 11.1.3 of Griffith's "Introduction to Electrodynamics 4Ed" appears magnetic dipole radiation, which results in the equation above. According to the resultant equation, there is no magnetic field in the axis of the wire loop because theta=0. However, I think the magnetic flux density is at maximum value there although its time-varying due to the alternating current. What am I missing now? Thanks for your concerns in advance.
For a time-independent current loop, the B-field is strongest for ##\theta = 0## (for a given ##r##), as you are thinking. But recall that the field falls off rapidly with distance as ##1/r^3##.

In Griffiths' calculation in section 11.1.3, he is considering a radiating, harmonically oscillating current loop. In this case, you get "radiation terms" in the results for B and E that decrease much more slowly with distance as ##1/r##. So for "large ##r##", only the radiation terms are significant. Note, in particular, Griffiths' "approximation 3" given as relation (11.34) on page 475; namely, assume ##r \gg c/\omega##.

It would probably be helpful for you to work problem 11.5 on page 477 (at least the first part where you are asked to find the fields without making approximation 3.)
 
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Related to On the magnetic dipole radiation in Griffith's book

What is magnetic dipole radiation?

Magnetic dipole radiation refers to the electromagnetic radiation emitted by a system when its magnetic dipole moment oscillates. It is analogous to electric dipole radiation, but it involves the magnetic field components instead of the electric field components.

How does Griffiths describe the magnetic dipole moment in the context of radiation?

In Griffiths' book, the magnetic dipole moment is described as a vector quantity that represents the strength and orientation of a magnetic source. For radiation, it typically involves time-dependent oscillations of this moment, which produce varying magnetic fields and consequently, electromagnetic radiation.

What are the key differences between electric dipole and magnetic dipole radiation?

The key differences lie in the source of the radiation and the nature of the fields involved. Electric dipole radiation arises from oscillating electric dipoles and primarily involves the electric field. Magnetic dipole radiation, on the other hand, originates from oscillating magnetic dipoles and involves the magnetic field. Additionally, the angular distribution and polarization of the emitted radiation differ between the two types.

What mathematical expressions are used to describe magnetic dipole radiation in Griffiths' book?

Griffiths uses several mathematical expressions to describe magnetic dipole radiation, including the vector potential (\( \mathbf{A} \)), the magnetic dipole moment (\( \mathbf{m} \)), and the far-field approximations. The radiation fields are derived using these quantities, often employing the retarded potentials and multipole expansion techniques.

How does Griffiths approach the derivation of the radiation fields for a magnetic dipole?

Griffiths approaches the derivation by first considering the vector potential produced by an oscillating magnetic dipole. He then uses the relationship between the vector potential and the electromagnetic fields to derive the expressions for the electric and magnetic fields in the far-field region. The derivation involves the use of spherical coordinates and the application of boundary conditions appropriate for radiation problems.

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