Q143 / 180 · JIPMER 2018 · Physics

Electron revolving with speed v is producing magnetic field B at centre. Find relation between radius of path $B$ and $v$ ?

  • A
    $B \propto V \propto \frac{1}{r}$
  • Correct answer
    B
    $B \propto V \propto \frac{1}{r^2}$
  • C
    $B \propto V^2 \propto \frac{1}{r}$
  • D
    $B \propto v^2 \propto \frac{1}{r^2}$

Solution

$\because$ Magnetic field at center, $B=\frac{\mu_0 \mathrm{I}}{2 \mathrm{r}}$ where, $I=\frac{\mathrm{e}}{\mathrm{t}}$ or $\quad \mathrm{I}=\frac{\mathrm{e}}{\left(\frac{2 \pi \mathrm{r}}{\mathrm{v}}\right)}=\frac{\mathrm{ev}}{2 \pi \mathrm{r}}$ Hence, $B=\left(\frac{\mu_0}{2 \mathrm{r}}\right)\left(\frac{\mathrm{ev}}{2 \pi \mathrm{r}}\right)=\frac{\mu_0 \mathrm{ev}}{4 \pi \mathrm{r}^2}$ or $\quad B \propto \frac{V}{r^2}$

Hence, the correct answer is option: 'B'.

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