The product of angular speed and tangential s | vedclass

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Question

(C) The angular velocity $\omega_n$ is given by $\omega_n = \frac{v_n}{r_n}$.The product of angular velocity and tangential velocity is $P = \omega_n

Vedclass · Accepted Answer

Inversely proportional to $n^4$

Vedclass · Answer

(C) The angular velocity $\omega_n$ is given by $\omega_n = \frac{v_n}{r_n}$.The product of angular velocity and tangential velocity is $P = \omega_n 	imes v_n = \frac{v_n^2}{r_n}$.For a hydrogen atom,the velocity in the $n^{	ext{th}}$ orbit is $v_n = \frac{v_0}{n}$ and the radius is $r_n = n^2 r_0$,where $v_0$ and $r_0$ are constants.Substituting these values into the expression for $P$:$P = \frac{(v_0/n)^2}{n^2 r_0} = \frac{v_0^2}{n^2 \cdot n^2 r_0} = \frac{v_0^2}{r_0 n^4}$.Thus,the product is inversely proportional to $n^4$ (i.e.,$P \propto \frac{1}{n^4}$).

The product of angular speed and tangential speed of an electron in the $n^{\text{th}}$ orbit of a hydrogen atom is:

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