Chapter 16, Relativity Iv Video Solutions, Electromagnetic Fields and Waves: Including Electric Circuits

Problem 1

(16.1) The invariance of electric charge
Imagine that clectric charge is not invariant and that $Q=Q_{0}[1-$ $\left.\left(v^{2} / c^{2}\right)\right]^{1 / 2}$. (Remember that charge is, in fact, invariant, according to all experiments performed to date). The charge $Q_{0}$ is that measured by an observer moving with the charge, and $Q$ is the charge for an observer moving at a velocity $q$ with respect to it.

If the electrons in a given sample have an average kinetic energy of 100 electronvolts, what percentage increase in their charge must we expect if their velocity increases by $1 \%$ ?

Dominador Tan

Numerade Educator

08:12

Problem 2

(16.2) Conduction and convection currents in a moving ring
A square conducting ring carries a current $I^{\prime}$ in its own reference frame, as in Fig. 16-11. Its cross section is $A^{\prime}$.
(a) The ring moves at a velocity $\boldsymbol{V}$ in the direction normal to its plane.
Find the current and the charge density with respect to a fixed reference frame.
(b) The ring moves to the right at a velocity $V \hat{t}$.
Find the currents and the charge densities in a fixed reference frame.
(c) The motion of the space charge at a velocity $V \hat{x}$ gives a convection current. Calculate the convection and conduction currents in the four sides.

Linda Winkler

Numerade Educator

04:07

Problem 3

(16.5.3) Alternate expressions for $\boldsymbol{E}$ and $\boldsymbol{B}$
Show that
$$
\boldsymbol{E}=\frac{\gamma Q r}{4 \pi \epsilon_{0}\left(\gamma^{2} x^{2}+y^{2}+z^{2}\right)^{12}}, \quad B=\frac{\mu_{0} \gamma Q \boldsymbol{V} \times r}{4 \pi\left(\gamma^{2} x^{2}+y^{2}+z^{2}\right)^{2 / 2}}
$$

Uma Kumari

Numerade Educator

04:26

Problem 4

( 16.5.3) The field of a 10 -megaelectronvolt proton
Plot $E$ and $B$ as functions of the time at a point $P$ one centimeter away from the path of a 10 megaelectronvolt proton. Set $P$ at $(0.0 .01,0)$, with the charge at $\left(\boldsymbol{V}_{t}, 0,0\right)$.

Dading Chen

Numerade Educator

02:10

Problem 5

(16.5) The force between electrons moving side by side Calculate the force, as observed in the laboratory, between two electrons
moving side by side along parallel paths 1 millimeter apart if they each have a kinetic energy of (a) 1 electronvolt and (b) 1 megaelectronvolt. Use Table $15-4$

Vishal Gupta

Numerade Educator

01:44

Problem 6

(16.6) $\boldsymbol{E} \cdot \boldsymbol{B}$ is invariant
Show that $\boldsymbol{E} \cdot \boldsymbol{B}$ is invariant under a Lorentz transformation.

Manik Pulyani

Numerade Educator

08:08

Problem 7

(16.6) $B^{2}-E^{2} / c^{2}$ is invariant
Show that $B^{2}-E^{2} / c^{2}$ is invariant under a Lorentz transformation.

Alexander Lorenzo

Numerade Educator

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Problem 8

The angle between $\boldsymbol{E}$ and $\boldsymbol{B}$ is not invariant Show that the angle between $\boldsymbol{E}$ and $\boldsymbol{B}$ is not invariant.

Hoan Nguyen

Numerade Educator

01:57

Problem 9

$(16.6)$ Transformation of a relative permittivity $\epsilon$,
A dielectric-filled parallel-plate capacitor moves at a velocity $\mathscr{V} \hat{x}$ with its plates (a) parallel to the $x z$ plane (Fig. 16-10) and (b) parallel to the $y z$ plane. Show that $\epsilon_{r}=\epsilon_{r}^{\prime}$ in both cases.

Mahendra Kumar

Numerade Educator

04:47

Problem 10

(16.6) The transformation of $P^{\prime}$
A dielectric situated in frame $S^{\prime}$ contains $N^{*}$ atoms per cubic meter, each atom possessing a dipole moment $p^{\prime}=Q s^{\prime} .$ So $P^{\prime}=N^{\prime \prime} Q s^{\prime}$.
Show that, with respect to frame $S, P=P_{\|}^{\prime}+\gamma P_{1}^{\prime}$

Khoobchandra Agrawal

Numerade Educator

02:02

Problem 11

(16.6) Time-independent magnetic fiEld.
In reference frame $S$ we have a constant magnetic field and no electric fiEld.

Show that $\boldsymbol{E}^{\prime}=\boldsymbol{V} \times \boldsymbol{B}^{\prime}$ in $S^{\prime}$. Note the prime on the right-hand side. So $\boldsymbol{E}^{\prime}$ is perpendicular to both $\boldsymbol{V}$ and $\boldsymbol{B}^{\prime}$.

Dominador Tan

Numerade Educator