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## Homework Statement

I reference problem 9.10 Purcell's Electricity and Magnetism (3rd ed).

A very thin straight wire carries current ##I## from infinity radially inward onto a conducting shell with radius R. Show that the total flux of the Poynting vector away from an imaginary tube of radius ##b## surrounding the wire equals the rate of change of the electric field.

I'm having trouble with several key ideas necessary to solve this problem, any assistance is greatly appreciated. I will write my understanding of the problem below to illustrate the issues.

## Homework Equations

## The Attempt at a Solution

There are 2 fields coming from the wire. There is a ##\vec{B} _w## looping around the wire and an ##\vec{E} _w## emanating radially from the wire, as if it were from a static line of charge.

There are also 2 fields associated with the sphere. There is an ##\vec{E} _s (t)## pointing radially out from the sphere, which induces a ##\vec{B} _s##, which has a radially -pointing (wrt sphere) curl as given by

$$\nabla \times \vec{B} _s = \mu _0 \epsilon _0 \ \partial _t \vec{E} _s (t)$$

To me, the Poynting vector evaluated at the surface of the tube should be in the form

$$\vec{S} = \frac{1}{\mu _0} \big ( [\vec{E} _w + \vec{E} _s (t)] \times [\vec{B} _w + \vec{B} _s ] \big )$$

However, the solutions manual states that

$$\vec{S} = \frac{1}{\mu _0} \big ( \vec{E} _s (t) \times \vec{B} _w \big )$$

I can sort of understand why ##\vec{E} _w## was omitted; the resulting Poynting vector term points along the wire, that is to say energy (associated with the current) travels towards the sphere instead of going "outwards" from the imaginary tube. However, I can't understand why the ##\vec{B} _s ## term was omitted as well. Should we not be evaluating the total fields at the points of interest?

Also, let's say I find that the ##\nabla \times \vec{B} _s## is independent of time, am I then allowed to say that ## \vec{B} _s## is independent of time as well, and so the flux of the Poynting vector does not go into increasing the energy density of the ##\vec{B} _s## field?

I apologize in advance for my slightly confused phrasing. Many thanks in advance for any assistance.