Question

Consider Bohr's model for hydrogen atom. Magnetic field due to single electron at centre of atom depends on $p^{th}$ power of electronic charge as $B=Ke^p$. Here $K$ being some factor with proper unit. Find the value of $p$.

Answer 1

7

Answer 2

In Bohr's model, the magnetic field ($B$) at the center of the hydrogen atom due to the orbiting electron is proportional to $\frac{ev}{r^2}$, where $e$ is the electronic charge, $v$ is the orbital speed, and $r$ is the orbital radius. From Bohr's postulates, we derive the dependencies of $v$ and $r$ on $e$:

1. The force balance equation $\frac{mv^2}{r} = \frac{1}{4\pi\epsilon_0} \frac{e^2}{r^2}$ implies $v^2 \propto \frac{e^2}{mr}$, so $v \propto \frac{e}{\sqrt{mr}}$.
2. The quantization of angular momentum $mvr = \frac{nh}{2\pi}$ implies $r \propto \frac{1}{mv}$. Combining these, we find that $r \propto e^{-2}$ and $v \propto e^2$. Substituting these dependencies into the expression for $B$: $B \propto e \cdot v \cdot r^{-2}$ $B \propto e \cdot (e^2) \cdot (e^{-2})^{-2}$ $B \propto e \cdot e^2 \cdot e^4$ $B \propto e^{1+2+4}$ $B \propto e^7$ Therefore, $p=7$.