184_notes:examples:week10_radius_motion_b_field

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184_notes:examples:week10_radius_motion_b_field [2017/11/02 21:21] – dmcpadden184_notes:examples:week10_radius_motion_b_field [2018/07/03 14:01] (current) – curdemma
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 +[[184_notes:q_path|Return to Path of a Charge through a Magnetic Field notes]]
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 =====Radius of Circular Motion in a Magnetic Field===== =====Radius of Circular Motion in a Magnetic Field=====
 Suppose you have a moving charge $q>0$ in a magnetic field $\vec{B} = -B \hat{z}$. The charge has a velocity of $\vec{v} = v\hat{x}$, and a mass $m$. What does the motion of the charge look like? What if the charge enters the field from a region with $0$ magnetic field? Suppose you have a moving charge $q>0$ in a magnetic field $\vec{B} = -B \hat{z}$. The charge has a velocity of $\vec{v} = v\hat{x}$, and a mass $m$. What does the motion of the charge look like? What if the charge enters the field from a region with $0$ magnetic field?
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   * We represent the two situations below.   * We represent the two situations below.
  
-{{ 184_notes:10_circular_setup.png?600 |Moving Charge in a Magnetic Field}}+[{{ 184_notes:10_circular_setup.png?600 |Moving Charge in a Magnetic Field}}]
  
 ====Solution==== ====Solution====
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 $$\vec{F}= q (v\hat{x}) \times (-B\hat{z}) = qvB\hat{y}$$ $$\vec{F}= q (v\hat{x}) \times (-B\hat{z}) = qvB\hat{y}$$
  
-{{ 184_notes:10_circular_force.png?600 |Force on the Moving Charge}}+[{{ 184_notes:10_circular_force.png?600 |Force on the Moving Charge}}]
  
 So, the force on the charge is at first perpendicular to its motion. This is pictured above. You can imagine that as the charge's velocity is directed a little towards the $\hat{y}$ direction, the force on it will also change a little, since the cross product that depends on velocity will change a little. In fact, if you remember from [[184_notes:q_path|the notes]], this results in circular motion if the charge is in a constant magnetic field. So, the force on the charge is at first perpendicular to its motion. This is pictured above. You can imagine that as the charge's velocity is directed a little towards the $\hat{y}$ direction, the force on it will also change a little, since the cross product that depends on velocity will change a little. In fact, if you remember from [[184_notes:q_path|the notes]], this results in circular motion if the charge is in a constant magnetic field.
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 So now we know the radius. You can imagine that if the particle entered from outside the region of magnetic field, it would take the path of a semicircle before exiting the field in the opposite direction. Below, we show what the motion of the particle would be in each situation. So now we know the radius. You can imagine that if the particle entered from outside the region of magnetic field, it would take the path of a semicircle before exiting the field in the opposite direction. Below, we show what the motion of the particle would be in each situation.
  
-{{ 184_notes:10_circular_motion.png?700 |Motion of the Moving Charge}}+[{{ 184_notes:10_circular_motion.png?700 |Motion of the Moving Charge}}]
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  • Last modified: 2017/11/02 21:21
  • by dmcpadden