Fyneman exercise II ch1-5

Q.

In the region of space of interest there is a uniform magnetic field, such that $B_x=0$, $B_y=0$, $B_z=B_0$. The field is constant in time, and there are no currents or electric fields in the region of space we consider.

A particle of mass $m$ and positive charge $q$ is started at $x=0$, $y=0$, $z=0$ with a velocity $v$ in the $+x$ direction. Sketch and describe quantitatively in terms of $B_0$, $m$, $v$, and $q$ the path of the particle. (Assume ${v \over c} << 1$)

Suppose, that $B_x=0$, $B_y=0$, but $B_z=B_0+ax$. For $(ax)$ always small compared to $B_0$, but not completely negligible, show on a sketch the qualitative behavior of the particle trajectory. (See Charpak, et al, Physical Review Letters, Vol. 6, 128 (1961) for the use of a similar field in an important expriment.) Show that the field just postulated is inconsistent with one of Maxwell’s equations if the field fills a finite volume of space and, as above, you assume there are no currents or electric fields in the volume.

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Trial.

${d \over dt}\left[ {m\vec{v} \over (1-{v^2 \over c^2})^{1 \over 2}}\right]=q(\vec{E}+\vec{v}\times\vec{B})\tag{1}$

For $\vec{E}=0$, ${v \over c}<<1$, $\vec{v}=v\hat{e}_x$,

${d\vec{v(t)} \over dt}=-{qvB_0\over m}\hat{e}_y\tag{2}$

$v_y(t)=-{qvB_0 \over m}t+C\tag{3}$

From Initial condition, $v_y(0)=C=0$,

$v_y(t)=-{qvB_0 \over m}t\tag{4}$

$\therefore \vec{v}=v\hat{e}_x-{qvB_0 \over m}t\hat{e}_y\tag{5}$

$x(t)=vt$, $y(t)=-{qvB_0 \over 2m}t^2$

$y(x)=-{qB_0 \over 2vm}x^2\tag{6}$

Trajectory is hyperbola.

If $B_z = B_0+ax$, from eq.(2),

${d\vec{v}(t) \over dt}=-{qv(B_0+avt) \over m}\hat{e}_y\tag{7}$

$v_y(t)=-{qv(B_0t+{1\over2}avt^2) \over m}\tag{8}$

$x(t)=vt$, $y(t)=-{qv(B_0t^2+{1\over3}avt^3) \over 2m}t$

$y(x)=-{q \over 2vm}(B_0x^2+{1\over3}ax^3)\tag{9}$

One of Maxwell’s equation is
$\nabla \times \vec{E}=-{\partial \vec{B} \over \partial t}\tag{10}$

Since $B_z$ is $B_0+avt$, $\nabla \times \vec{E}\ne 0$.
Therefore postulation of no electric field is inconsistent!