Question
An electron moves in a circular orbit with a uniform speed $$v.$$ It produces a magnetic field $$B$$ at the centre of the circle. The radius of the circle is proportional to
A.
$$\frac{B}{v}$$
B.
$$\frac{v}{B}$$
C.
$$\sqrt {\frac{v}{B}} $$
D.
$$\sqrt {\frac{B}{v}} $$
Answer :
$$\sqrt {\frac{v}{B}} $$
Solution :
The time period of electron moving in a circular orbit,
$$T = \frac{{{\text{Circumference of circular path}}}}{{{\text{Speed}}}} = \frac{{2\pi r}}{v}$$
Now, equivalent current due to flow of electron is given by
$$i = \frac{q}{T} = \frac{e}{{\left( {\frac{{2\pi r}}{v}} \right)}} = \frac{{ev}}{{2\pi r}}\,\,\left[ {q = e} \right]$$
Magnetic field at centre of circle
$$\eqalign{
& B = \frac{{{\mu _0}i}}{{2r}} = \frac{{{\mu _0}ev}}{{4\pi {r^2}}}\,\,\,\left( {i = \frac{{eV}}{{2\pi r}}} \right) \cr
& \Rightarrow r \propto \sqrt {\frac{V}{B}} \cr} $$