An electron enters with a velocity ${\rm{\vec v}},{{\rm{v}}_0}{\rm{\hat i}}$ 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 $\mathrm{xy}$ - plane. Suggest a configuration of fields $\mathrm{E}$ and $\mathrm{B}$ that can lead to it.
An electron enters with a velocity ${\rm{\vec v}},{{\rm{v}}_0}{\rm{\hat i}}$ 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 $\mathrm{xy}$ - plane. Suggest a configuration of fields $\mathrm{E}$ and $\mathrm{B}$ that can lead to it.
Due to magnetic field, electron is moving on circular path in $x y$-plane linear speed of electron is increase by increasing electric field in $x$-direction, so that radius of trajectory of path also increases.
Thus, electron is moving on helical path.
Change magnetic field as, $\overrightarrow{\mathrm{B}}=\mathrm{B}_{0} \hat{k}$
Now velocity of electron, $\vec{v}=v_{0} \hat{i}$
Magnetic force on electron when it enter in magnetic field,
$\overrightarrow{\mathrm{F}}=-e\left(v_{0} \hat{i} \times \mathrm{B}_{0} \hat{k}\right)$
$=-e v_{0} \mathrm{~B}_{0}(\hat{i} \times \hat{k})$
$=-e v_{0} \mathrm{~B}_{0}(-\hat{j})$
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- [IIT 2004]
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