In a region of empty space there is a uniform electric field with and a uniform…

PHYSICS · Multi-part · Fermi University Physics · Electricity, Magnetism & Electromagnetism · Relativity & Gravitation · Relativistic dynamics and fields in flat spacetime

The Geometry of Gravity, Ch The stress-energy tensor and relativistic continua

Problem

In a region of empty space there is a uniform electric field

\vec E=E_0\,\hat{\mathbf{y}}

with

E_0=3.0\times10^5\ \mathrm{V\,m^{-1}}

and a uniform magnetic field

\vec B=B_0\,\hat{\mathbf{z}}

with

B_0=2.0\times10^{-3}\ \mathrm{T}

. Use

T^{\mu\nu}_{\rm EM}=\mu_0^{-1}\left(F^{\mu\alpha}F^\nu{}_\alpha-\tfrac14\eta^{\mu\nu}F^{\alpha\beta}F_{\alpha\beta}\right)

with

F_{0i}=-E_i/c

and

F_{ij}=\epsilon_{ijk}B_k

, and take

\epsilon_0=8.85\times10^{-12}\ \mathrm{F\,m^{-1}}

\mu_0=1.26\times10^{-6}\ \mathrm{H\,m^{-1}}

and

c=3.00\times10^8\ \mathrm{m\,s^{-1}}

. ((a)) Show that

F^{\alpha\beta}F_{\alpha\beta}=2\left(|\vec B|^2-|\vec E|^2/c^2\right)

for any fields, and show that the trace

T^\mu{}_\mu

vanishes whatever

\vec E

and

\vec B

are. ((b)) Show that

T^{0x}_{\rm EM}=(\vec E\times\vec B)_x/(\mu_0c)

, so that

cT^{0x}

is the

x

component of the Poynting vector. ((c)) Evaluate

T^{00}

and

T^{0x}

for the given fields, and the speed

cT^{0x}/T^{00}

at which energy is carried along

x

. Say how the result sits with the dominant energy condition.

Related theory

Relativistic electrodynamics

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