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Problem 20 Medium Difficulty

A particle of charge $q$ and mass $m$, moving with a constant speed $v$, perpendicular to a constant magnetic field $B$, follows a circular path. If in this case the angular momentum about the center of this circle is quantized so that $m v r=2 n \hbar,$ show that the allowed radii for the particle are
$$r_{n}=\sqrt{\frac{2 n \hbar}{q B}}$$

Answer

$r_{n}=\sqrt{\frac{2 n h}{q B}}$

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Top Physics 103 Educators
Elyse G.

Cornell University

Christina K.

Rutgers, The State University of New Jersey

Zachary M.

Hope College

Aspen F.

University of Sheffield

Video Transcript

In this exercise, we have a particle of charge Q and mass M that moves in a circular motion and a magnetic field be You also have that The angular momentum is Kwan Tai such that l is equal to two an H bar and are going. This exercise is to prove that our is equal to the square root off to an H bar over Q. Be okay. So in the first step, I'm going to start with the first step here, which is to find are the radius and the function of the mass, the speed, the magnetic feud and the charge peak que Okay, So in order to do that, I'm gonna take this interpretive force and B squared over R and make it equal to the magnetic force, which is just people to be times the charge Q Times the speed V of the particle. So we have that M V over B. Here is people to our okay. And from here we get that V is equal to me. Que are over him. And I'm gonna save this equation here because we're going to use it, okay. And the second stop step no is to remember that l The end of momentum is equal to two and h bar and the angular momentum could be redness and VR. Okay, this is to an h bar, and I'm gonna substitute V for the equation for V that we found in the step one that IHS m times V, which is me que are over him. Times are equals two in H bar. So the masses cancel out and you get be que are square equals two and h bar. So our is equal to the square root of two NH bar over key. You be okay. And this is what we wanted to show. So our exercises complete.

Universidade de Sao Paulo
Top Physics 103 Educators
Elyse G.

Cornell University

Christina K.

Rutgers, The State University of New Jersey

Zachary M.

Hope College

Aspen F.

University of Sheffield