Energy Flow Through Empty Space: Poynting's Hidden Highway · AI Deep Generator #DEEP-FC3FC806

Physics · electromagnetism · modern-physics

Problem

A coaxial cable carries steady current

I_0

from a battery of voltage

V_0

. The inner conductor has radius

a

, the outer conductor has inner radius

b

, and the space between them is vacuum. The cable has length

L \gg b

and the conductors have negligible resistance. (a) [8 pt] Find the electric field

\vec{E}(r)

and magnetic field

\vec{B}(r)

in the region

a < r < b

between the conductors. Express your answer in cylindrical coordinates

(r, \phi, z)

with

z

along the cable axis. (b) [10 pt] Calculate the Poynting vector

\vec{S} = \frac{1}{\mu_0}\vec{E} \times \vec{B}

in this region. Show that the total electromagnetic power flow through any cross-section equals

V_0 I_0

. (c) [12 pt] The electromagnetic field carries momentum density

\vec{g} = \epsilon_0 \vec{E} \times \vec{B}

. Find the total electromagnetic momentum stored in the cable. Express your result in terms of the cable's inductance

L_{\text{cable}} = \frac{\mu_0 L}{2\pi} \ln\left(\frac{b}{a}\right)

. (d) [10 pt] At

t = 0

, the circuit is suddenly broken at one end. During the transient, the current decreases as

I(t) = I_0 e^{-t/\tau}

where

\tau = L_{\text{cable}}/R_{\text{load}}

and

R_{\text{load}}

is the load resistance. Find the rate of change of electromagnetic momentum in the cable. (e) [12 pt] During the current decay, electromagnetic waves are radiated from the cable ends. Using momentum conservation, find the net radiation pressure force on the cable system. Explain the physical origin of this force. (f) [12 pt] The radiated power can be estimated as

P_{\text{rad}} = \frac{\mu_0 c}{6\pi}\left(\frac{dI}{dt}\right)^2 A_{\text{eff}}

where

A_{\text{eff}} = \pi(b^2 - a^2)

is the effective radiating area. Find the total energy radiated during the current decay. (g) [12 pt] Compare the radiated energy to the initial stored magnetic energy

U_B = \frac{1}{2}L_{\text{cable}}I_0^2

. Show that for

R_{\text{load}} \gg \sqrt{\frac{\mu_0 c A_{\text{eff}}}{3\pi L_{\text{cable}}}}

, radiation is negligible. (h) [14 pt] For a numerical example, consider

a = 1

mm,

b = 3

mm,

L = 10

I_0 = 1

A, and

R_{\text{load}} = 50

Ω. Calculate: (i) the initial electromagnetic momentum, (ii) the radiation recoil force, and (iii) the fraction of energy radiated. Comment on why coaxial cables are excellent for containing electromagnetic energy.

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