91Ó°ÊÓ

A charge$\mathit{q}$ is released from rest at the origin, in the presence of a uniform electric field$\mathit{E}\mathbf{=}{\mathbf{E}}_{\mathbf{0}}\widehat{\mathbf{z}}$ and a uniform magnetic field$\mathit{B}\mathbf{=}{\mathbf{B}}_{\mathbf{0}}\widehat{\mathbf{x}}$ . Determine the trajectory of the particle by transforming to a system in Which,$\mathit{E}\mathbf{=}\mathbf{0}$ , finding the path in that system and then transforming back to the original system. Assume${\mathit{E}}_{\mathbf{0}}\mathbf{<}\mathit{c}{\mathit{B}}_{\mathbf{0}}$ .Compare your result with Ex. 5.2.

Use the Larmor formula (Eq. 11.70) and special relativity to derive the Lienard formula (Eq. 11. 73).

$\begin{array}{l}\mathbf{P}\mathbf{=}\frac{{\mathbf{μ}}_{\mathbf{0}}{\mathbf{q}}^{\mathbf{2}}{\mathbf{a}}^{\mathbf{2}}}{\mathbf{6}\mathbf{Ï€}\mathbf{c}}\mathbf{\text{ â¶Ä‰â¶Ä‰}}\mathbf{(}\mathbf{11}\mathbf{.}\mathbf{70}\mathbf{)}\\ \mathbf{P}\mathbf{=}\frac{{\mathbf{μ}}_{\mathbf{0}}{\mathbf{q}}^{\mathbf{2}}{\mathbf{γ}}^{\mathbf{6}}}{\mathbf{6}\mathbf{Ï€}\mathbf{c}}\mathbf{(}{\mathbf{a}}^{\mathbf{2}}\mathbf{-}{\mathbf{\left|}\frac{\mathbf{Ï\dots }\mathbf{×}\mathbf{a}}{\mathbf{c}}\mathbf{\right|}}^{\mathbf{2}}\mathbf{)}\mathbf{\text{ â¶Ä‰â¶Ä‰}}\mathbf{(}\mathbf{11}\mathbf{.}\mathbf{73}\mathbf{)}\end{array}$

Sophie Zabar, clairvoyante, cried out in pain at precisely the instant her twin brother, 500km away, hit his thumb with a hammer. A skeptical scientist observed both events (brother’s accident, Sophie’s cry) from an airplane traveling at$\frac{\mathbf{12}}{\mathbf{13}}\mathit{c}$ to the right (Fig. 12.19). Which event occurred first, according to the scientist? How much earlier was it, in seconds?

(a) Write out the matrix that describes a Galilean transformation (Eq. 12.12).

(b) Write out the matrix describing a Lorentz transformation along the yaxis.

(c) Find the matrix describing a Lorentz transformation with velocity v along the x axis followed by a Lorentz transformation with velocity $\overline{\mathbf{v}}$along they axis. Does it matter in what order the transformations are carried out?

(a) Draw a space-time diagram representing a game of catch (or a conversation) between two people at rest, apart. How is it possible for them to communicate, given that their separation is spacelike?

(b) There's an old limerick that runs as follows:

There once was a girl named Ms. Bright,

Who could travel much faster than light.

She departed one day,

The Einsteinian way,

And returned on the previous night.

What do you think? Even if she could travel faster than light, could she return before she set out? Could she arrive at some intermediate destination before she set out? Draw a space-time diagram representing this trip.

Find the invariant product of the 4-velocity with itself, ${\mathit{η}}^{\mathbf{μ}}{\mathit{η}}^{\mathbf{μ}}$. Is localid="1654516875655" ${\mathit{η}}^{\mathbf{μ}}$timelike, spacelike, or lightlike?

Consider a particle in hyperbolic motion,

$\mathbf{x}\left(t\right)\mathbf{=}\sqrt{{\mathbf{b}}^{\mathbf{2}}\mathbf{+}{\left(\mathrm{ct}\right)}^{\mathbf{2}}}\mathbf{,}\mathbf{y}\mathbf{=}\mathbf{z}\mathbf{=}\mathbf{0}$

(a) Find the proper time role="math" localid="1654682576730" $\mathit{Ï„}$as a function of $\mathit{Ï„}$, assuming the clocks are set so that$\mathit{Ï„}\mathbf{=}\mathbf{0}$ when$\mathit{Ï„}\mathbf{=}\mathbf{0}$ . [Hint: Integrate Eq. 12.37.]

(b) Find x and v (ordinary velocity) as functions of$\mathit{Ï„}$ .

(c) Find${\mathit{η}}^{\mathbf{μ}}$ (proper velocity) as a function of $\mathit{τ}$.

In a pair annihilation experiment, an electron (mass m) with momentum ${\mathbf{p}}_{\mathbf{6}}$hits a positron (same mass, but opposite charge) at rest. They annihilate, producing two photons. (Why couldn’t they produce just one photon?) If one of the photons emerges at $\mathbf{60}\mathbf{°}$to the incident electron direction, what is its energy?

Define proper acceleration in the obvious way:

${\mathit{Î\pm }}^{\mathbf{μ}}\mathbf{=}\frac{\mathbf{d}{\mathbf{η}}^{\mathbf{μ}}}{\mathbf{d}\mathbf{Ï„}}\mathbf{=}\frac{{\mathbf{d}}^{\mathbf{2}}{\mathbf{x}}^{\mathbf{μ}}}{\mathbf{d}{\mathbf{Ï„}}^{\mathbf{2}}}$

(a) Find${\mathit{Î\pm }}^{\mathbf{0}}$and α in terms of u and a (the ordinary acceleration).

(b) Express${\mathit{Î\pm }}_{\mathbf{μ}}{\mathit{Î\pm }}^{\mathbf{μ}}$in terms of u and a.

(c) Show that${\mathit{η}}^{\mathbf{μ}}{\mathit{Î\pm }}_{\mathbf{μ}}\mathbf{=}\mathbf{0}$.

(d) Write the Minkowski version of Newton’s second law, in terms of${\mathit{Î\pm }}_{\mathbf{μ}}$. Evaluate the invariant product${\mathit{K}}^{\mathbf{μ}}{\mathit{η}}_{\mathbf{μ}}$.

(a) Show that $\mathbf{(}\mathbf{E}\mathbf{â‹\dots }\mathbf{B}\mathbf{)}$is relativistically invariant.

(b) Show that $\mathbf{(}{\mathbf{E}}^{\mathbf{2}}\mathbf{-}{\mathbf{c}}^{\mathbf{2}}{\mathbf{B}}^{\mathbf{2}}\mathbf{)}$is relativistically invariant.

(c) Suppose that in one inertial system$\mathbf{B}\mathbf{=}\mathbf{0}$but $\mathbf{E}\mathbf{≠}\mathbf{0}$(at some point P). Is it possible to find another system in which the electric field is zero atP?