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Using your knowledge of circular (centripetal) motion, derive an equation for the radius r of the circular path that electrons follow in terms of the magnetic field B, the electrons' velocity v, charge e, and mass m. You may assume that the electrons move at right angles to the magnetic field.

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Final answer:

The radius of the circular path that electrons follow in a magnetic field is derived by equating the magnetic force to centripetal force, leading to the equation r = mv/(qB).

Step-by-step explanation:

To obtain the radius r of the circular path followed by an electron moving in a magnetic field, we start by equating the magnetic force to the centripetal force required to keep the electron moving in a circle.

The magnetic force FB acting on a charge q moving with velocity v at a right angle to a magnetic field B is given by FB = qvB. To maintain circular motion, this force must equal the centripetal force, which is given by Fc = mv2/r, where m is the mass of the electron.

Setting these two forces equal gives us qvB = mv2/r. When we solve this equation for the radius r, we get r = mv/(qB). This formula relates the radius r of the circular path to the mass m, charge q, velocity v, and magnetic field B.

User Johannes Staehlin
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6 votes

Answer:


r=(m.v)/(q.B)

Step-by-step explanation:

  • For an electron moving with velocity v in a magnetic field B perpendicular to the direction of velocity.
  • As the electron enters the field it starts experiencing a force due to magnetic field which always acts perpendicular to the velocity of electron which is in accordance with the Lorentz force. This force
    F_B is given by the equation:


F_B=q.v.B...........................................(1)

where q is the charge on the particle.

  • Due to the this force the path of the electron becomes circular under the influence of the magnetic field.

Now as we know that the force acting on the particle moving in the circular path is a centripetal force which is given as:


F_c=m.r.\omega^2....................................(2)

where:

m = mass of the particle

r = radius of the particle


\omega= angular velocity of the particle

Also, the relation between the angular and the linear velocity is as :


\omega=(v)/(r)........................................(3)

From the eq. (2) & (3)


F_c=m.(v^2)/(r)...................................(4)

from eq. (1) & (4)


F_c=F_B


m.(v^2)/(r)=q.v.B


r=(m.v)/(q.B)

User Marko Kevac
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