魅影直播

(a) What鈥檚 the percent error introduced when you use Galileo鈥檚 rule, instead of Einstein鈥檚, with${\mathbf{v}}_{\mathbf{AB}}\mathbf{=}\mathbf{5}\mathbf{}\mathbf{mi}\mathbf{/}\mathbf{h}\mathbf{}\mathbf{and}\mathbf{}{\mathbf{v}}_{\mathbf{BC}}\mathbf{=}\mathbf{60}\mathbf{}\mathbf{mi}\mathbf{/}\mathbf{h}$and?

(b) Suppose you could run at half the speed of light down the corridor of a train going three-quarters the speed of light. What would your speed be relative to the ground?

(c) Prove, using Eq. 12.3, that if${\mathbf{v}}_{\mathbf{AB}}\mathbf{<}\mathbf{c}\mathbf{}\mathbf{and}\mathbf{}{\mathbf{v}}_{\mathbf{BC}}\mathbf{}\mathbf{<}\mathbf{c}\mathbf{}\mathbf{then}\mathbf{}{\mathbf{v}}_{\mathbf{AC}}\mathbf{<}\mathbf{c}$Interpret this result.

Show that

${\mathbf{k}}_{\mathbf{渭}}{\mathbf{k}}^{\mathbf{渭}}\mathbf{=}\frac{\mathbf{胃}\mathbf{鈥�}\left({\displaystyle \frac{u^2}{c^2}}\right){\mathbf{cos}}^{\mathbf{2}}}{\mathbf{1}\mathbf{-}{\displaystyle \frac{{\mathbf{u}}^{\mathbf{2}}}{{\mathbf{c}}^{\mathbf{2}}}}}$

Where$\mathbf{胃}$is the angle between u and F.

Show that the (ordinary) acceleration of a particle of mass m and charge q, moving at velocity u under the influence of electromagnetic fields E and B, is given by

$\mathbf{a}\mathbf{=}\frac{\mathbf{q}}{\mathbf{m}}\sqrt{\mathbf{1}\mathbf{鈥�}{\mathbf{u}}^{\mathbf{2}}\mathbf{/}{\mathbf{c}}^{\mathbf{2}}}\left[E+u脳B-\frac{1}{c^2}u\left(u鈥�E\right)\right]$

[Hint: Use Eq. 12.74.]

In system ${\mathit{S}}_{\mathbf{0}}$, a static uniform line charge$\mathit{位}$ coincides with thez axis.

(a) Write the electric field${\mathit{E}}_{\mathbf{0}}$ in Cartesian coordinates, for the point $\left(x_0,y_0,z_0\right)$.

(b) Use Eq. 12.109 to find the electric in S, which moves with speedv in the x direction with respect to ${\mathit{S}}_{\mathbf{0}}$. The field is still in terms of $\left(x_0,y_0,z_0\right)$; express it instead in terms of the coordinates$\left(x,y,z\right)$ in S. Finally, write E in terms of the vector S from the present location of the wire and the angle$\mathit{胃}$ between S and $\hat{\mathbf{x}}$. Does the field point away from the instantaneous location of the wire, like the field of a uniformly moving point charge?

(a) Show that $\mathbf{(}\mathbf{E}\mathbf{鈰�}\mathbf{B}\mathbf{)}$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 $\mathbf{E}\mathbf{鈮�}\mathbf{0}$(at some point P). Is it possible to find another system in which the electric field is zero atP?

Work out the remaining five parts to Eq. 12.118.

As the outlaws escape in their getaway car, which goes,$\frac{\mathbf{3}}{\mathbf{4}}\mathit{c}$the police officer fires a bullet from the pursuit car, which only goes$\frac{\mathbf{1}}{\mathbf{2}}\mathit{c}$(Fig. 12.3). The muzzle velocity of the bullet (relative to the gun)$\frac{\mathbf{1}}{\mathbf{3}}\mathit{c}$is. Does the bullet reach its target (a) according to Galileo, (b) according to Einstein?

Recall that a covariant 4-vector is obtained from a contravariant one by changing the sign of the zeroth component. The same goes for tensors: When you 鈥渓ower an index鈥� to make it covariant, you change the sign if that index is zero. Compute the tensor invariants

${\mathbf{F}}^{\mathbf{渭惫}}\mathbf{}{\mathbf{F}}_{\mathbf{渭惫}\mathbf{}\mathbf{,}}{\mathbf{G}}^{\mathbf{渭惫}}\mathbf{}{\mathbf{G}}_{{}^{\mathbf{渭惫}}}\mathbf{}\mathbf{and}\mathbf{}{\mathbf{F}}^{\mathbf{渭惫}}\mathbf{}{\mathbf{G}}_{{}^{\mathbf{渭惫}}}$

in terms of E and B. Compare Prob. 12.47.

A straight wire along thez-axis carries a charge density$\mathbf{位}$traveling in the +z direction at speed v. Construct the field tensor and the dual tensor at the point role="math" localid="1654331549769" $\left(x,0,0\right)$.

Show that the second equation in Eq. 12.127 can be expressed in terms of the field tensor ${\mathbf{F}}^{\mathbf{渭谓}}$as follows:

localid="1654746948628" $\frac{\mathbf{鈭�}{\mathbf{F}}_{\mathbf{渭谓}}}{\mathbf{鈭�}{\mathbf{x}}^{\mathbf{位}}}\mathbf{+}\frac{\mathbf{鈭�}{\mathbf{F}}_{\mathbf{谓位}}}{\mathbf{鈭�}{\mathbf{x}}^{\mathbf{渭}}}\mathbf{+}\frac{\mathbf{鈭�}{\mathbf{F}}_{\mathbf{位渭}}}{\mathbf{鈭�}{\mathbf{x}}^{\mathbf{谓}}}\mathbf{=}\mathbf{0}$