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An electron enters with a velocity $\mathrm{v}=\mathrm{v}_0 \hat{\mathrm{i}}$ into a cubical region (faces parallel to coordinate planes) in which there are uniform electric and magnetic fields. The orbit of the electron is found to spiral down inside the cube in plane parallel to the $x-y$ plane. Suggest a configuration of fields $\mathrm{E}$ and $\mathrm{B}$ that can lead to it.
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Let, magnetic field $\mathrm{B}=\mathrm{B}_0 \hat{\mathrm{k}}$ in $\mathrm{y}$-axis and moving electron enters with the velocity $v=v_0 \hat{i}$ into a cubical region along $x$-axes.
Then the force on electron, using magnetic Lorentz force, is
$$
\mathrm{F}=-\mathrm{e}\left(\mathrm{v}_0 \mathrm{i} \times \mathrm{B}_0 \hat{\mathrm{k}}\right)=\mathrm{ev}_0 \mathrm{~B}_0 \hat{\mathrm{i}}
$$
which revolves the electron in $x-y$ plane.
The electric force $\mathrm{F}=-\mathrm{eE}_0 \hat{\mathrm{k}}$ accelerates electron along zaxis which in turn increases the radius of circular path. Hence particle traversed on spiral path.
The magnetic field revolves the charge particle in uniform circular motion in $\mathrm{x}$-y plane and electric field along $\mathrm{x}-$ direction increases the speed, which in turn increases the radius of circular path and hence, particle traversed on spiral path.
Then the force on electron, using magnetic Lorentz force, is
$$
\mathrm{F}=-\mathrm{e}\left(\mathrm{v}_0 \mathrm{i} \times \mathrm{B}_0 \hat{\mathrm{k}}\right)=\mathrm{ev}_0 \mathrm{~B}_0 \hat{\mathrm{i}}
$$
which revolves the electron in $x-y$ plane.
The electric force $\mathrm{F}=-\mathrm{eE}_0 \hat{\mathrm{k}}$ accelerates electron along zaxis which in turn increases the radius of circular path. Hence particle traversed on spiral path.
The magnetic field revolves the charge particle in uniform circular motion in $\mathrm{x}$-y plane and electric field along $\mathrm{x}-$ direction increases the speed, which in turn increases the radius of circular path and hence, particle traversed on spiral path.
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