# Does Poynting's Theorem only involve external fields?

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• jcap
In summary, Poynting's Theorem states that the rate of energy transfer from a region of space is equal to the rate of work done on a charge distribution plus the energy flux leaving that region. The theorem also includes equations for Poynting's vector and field energy density. When considering the electromagnetic field, it includes the field from all sources, including the charge distribution itself. However, point charges may cause mathematical issues in this theorem.
jcap
Poynting's Theorem (https://en.wikipedia.org/wiki/Poynting's_theorem) says:

The rate of energy transfer (per unit volume) from a region of space equals the rate of work done on a charge distribution plus the energy flux leaving that region.
$$-\frac{\partial u}{\partial t}=\mathbf{J}\cdot\mathbf{E}+\nabla\cdot\mathbf{S}$$
Poynting's vector is given by
$$\mathbf{S} = \frac{1}{\mu_0} \mathbf{E} \times \mathbf{B}$$
Field energy density is given by
$$u = \frac{1}{2}\left(\epsilon_0 \mathbf{E}\cdot\mathbf{E} + \frac{1}{\mu_0}\mathbf{B}\cdot\mathbf{B}\right)$$
Does the EM field ##\mathbf{E}##,##\mathbf{B}## include the field from the charge distribution itself or is it only the external field due to external sources?

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It is the field from all sources.

Note that point charges cause mathematical problems here.

vanhees71

## 1. What is Poynting's Theorem?

Poynting's Theorem is a fundamental law in electromagnetism that describes the relationship between electric and magnetic fields in a given region of space. It states that the net flow of electromagnetic energy through a surface is equal to the rate of change of electromagnetic energy within that region.

## 2. Does Poynting's Theorem only involve external fields?

No, Poynting's Theorem applies to both external and internal fields. It takes into account all sources of electromagnetic energy, including those generated by external sources and those within the region of interest.

## 3. How does Poynting's Theorem relate to the conservation of energy?

Poynting's Theorem is a manifestation of the law of conservation of energy. It shows that the total amount of electromagnetic energy in a given region remains constant, as energy is neither created nor destroyed but only transferred between electric and magnetic fields.

## 4. Can Poynting's Theorem be applied to any type of electromagnetic wave?

Yes, Poynting's Theorem is a general law that applies to all types of electromagnetic waves, including radio waves, light waves, and X-rays.

## 5. What are some practical applications of Poynting's Theorem?

Poynting's Theorem is used in various fields, including electrical engineering, telecommunications, and optics. It helps in understanding the propagation of electromagnetic waves, designing antennas, and calculating the power transmitted in a given system. It is also essential in the development of technologies such as wireless power transfer and solar energy harvesting.

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