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184_notes:i_b_force [2018/05/15 16:44] – curdemma184_notes:i_b_force [2021/07/13 11:58] (current) – schram45
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 Section 20.2 in Matter and Interactions (4th edition) Section 20.2 in Matter and Interactions (4th edition)
  
-[[184_notes:b_summary|Next Page: Summary of Magnetic Fields/Forces]] +/*[[184_notes:b_summary|Next Page: Summary of Magnetic Fields/Forces]]*/
- +
-[[184_notes:b_shapes|Previous Page: Shapes of Wire and Magnetic Field]]+
  
 ===== Magnetic Force on a Current Carrying Wire ===== ===== Magnetic Force on a Current Carrying Wire =====
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 {{youtube>cz3Q22KW7fs?large}} {{youtube>cz3Q22KW7fs?large}}
-==== Force on a little chunk ====+===== Force on a little chunk =====
 If we think about a long straight wire with a //__steady state current__//, we can model this simply as many moving charges in a wire. When that wire is placed in an external magnetic field (from some other source - either another wire or permanent magnet), each of the moving charges would feel a force. Collectively, this results in a net force on the wire by the magnetic field.  If we think about a long straight wire with a //__steady state current__//, we can model this simply as many moving charges in a wire. When that wire is placed in an external magnetic field (from some other source - either another wire or permanent magnet), each of the moving charges would feel a force. Collectively, this results in a net force on the wire by the magnetic field. 
  
-{{  184_notes:week11_6.png?350}}+[{{  184_notes:week11_6.png?350|Current carrying wire in an external magnetic field}}]
  
 To calculate this magnetic force on the wire, we follow a very similar set of steps. We can start by dividing the wire into little chunks and only thinking about the amount of moving charge in that little chunk. Since we are interested in the small bit of force on the little bit of moving charge in that piece of the wire, we could write this as: To calculate this magnetic force on the wire, we follow a very similar set of steps. We can start by dividing the wire into little chunks and only thinking about the amount of moving charge in that little chunk. Since we are interested in the small bit of force on the little bit of moving charge in that piece of the wire, we could write this as:
 $$d\vec{F}= dq \vec{v}\times\vec{B}$$ $$d\vec{F}= dq \vec{v}\times\vec{B}$$
-where dq is the small amount of charge in the little piece of the wire, $\vec{v}$ is the speed with which that small bit of charge is moving, and $\vec{B}$ is the external magnetic field.+where $dq$ is the small amount of charge in the little piece of the wire, $\vec{v}$ is the speed with which that small bit of charge is moving (units of $\frac{m}{s}$, and $\vec{B}$ is the external magnetic field (units of $T$).
  
 However, especially for wires, it is often more useful to think about the current in that piece rather than the individual moving charges. So to rewrite this in terms of current, we start by rewriting the velocity in terms of the length and time ([[184_notes:b_current|just like we did before]]) $\vec{v}=\frac{d\vec{l}}{dt}$. So we get the force from the small piece of the wire to be: However, especially for wires, it is often more useful to think about the current in that piece rather than the individual moving charges. So to rewrite this in terms of current, we start by rewriting the velocity in terms of the length and time ([[184_notes:b_current|just like we did before]]) $\vec{v}=\frac{d\vec{l}}{dt}$. So we get the force from the small piece of the wire to be:
 $$d\vec{F}= dq \frac{d\vec{l}}{dt}\times\vec{B}$$ $$d\vec{F}= dq \frac{d\vec{l}}{dt}\times\vec{B}$$
  
-Then, since dt represents a small amount of time, dl represents a small amount of length, and dq represents a small amount of charge, we will treat these as independent and rewrite: +Then, since $dt$ represents a small amount of time, $dl$ represents a small amount of length, and $dq$ represents a small amount of charge, we will treat these as independent and rewrite: 
 $$dq \cdot \frac{d\vec{l}}{dt} =  \frac{dq \cdot d\vec{l}}{dt}=\frac{dq}{dt}\cdot d\vec{l}$$  $$dq \cdot \frac{d\vec{l}}{dt} =  \frac{dq \cdot d\vec{l}}{dt}=\frac{dq}{dt}\cdot d\vec{l}$$ 
  
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 using the fact that $\frac{dq}{dt}$ is the definition of conventional current (the amount of charge passing a point per second).  using the fact that $\frac{dq}{dt}$ is the definition of conventional current (the amount of charge passing a point per second). 
  
-{{  184_notes:week11_7.png?350}}+[{{  184_notes:week11_7.png?350|Magnetic Force, F, produced on a current carrying wire in an external magnetic field, B}}]
 This means that the small amount of force on the wire is given by: This means that the small amount of force on the wire is given by:
 $$d\vec{F}= I d\vec{l} \times \vec{B}$$ $$d\vec{F}= I d\vec{l} \times \vec{B}$$
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 Note: that the force is still given by the cross product between the $d\vec{l}$ and the $\vec{B}$, so the force on the piece of wire is //still// perpendicular to both the direction of the moving charges ($d\vec{l}$) and perpendicular to the magnetic field ($\vec{B}$). This means we can still use the [[184_notes:rhr|right hand rule]] to figure out the direction of the $d\vec{F}$. Note: that the force is still given by the cross product between the $d\vec{l}$ and the $\vec{B}$, so the force on the piece of wire is //still// perpendicular to both the direction of the moving charges ($d\vec{l}$) and perpendicular to the magnetic field ($\vec{B}$). This means we can still use the [[184_notes:rhr|right hand rule]] to figure out the direction of the $d\vec{F}$.
    
-==== Force on the whole wire ====  +===== Force on the whole wire ===== 
-Now that we have the magnetic force on a small piece of the wire, we can find the total force on the wire from the external magnetic field by adding up the contributions from each little piece of the wire. Since we have the small bit of force from the small bit of wire, we will add these using a integral:+Now that we have the magnetic force on a small piece of the wire, we can find the total force on the wire from the external magnetic field by adding up the contributions from each little piece of the wire. Since we have the small bit of force from the small bit of wire, we will add these using an integral:
 $$\vec{F}_{wire}= \int_{wire} d\vec{F} = \int_{l_i}^{l_f} I d\vec{l} \times \vec{B}$$ $$\vec{F}_{wire}= \int_{wire} d\vec{F} = \int_{l_i}^{l_f} I d\vec{l} \times \vec{B}$$
 Here we want to pick the limits of the integral to be from the starting point of the wire ($l_i$) to the end of the wire ($l_f$) so we are adding up over the whole length of the wire. This form of the force will //always// work to find the magnetic force on the whole wire - we have not made very many assumptions so far in coming up with this equation.  Here we want to pick the limits of the integral to be from the starting point of the wire ($l_i$) to the end of the wire ($l_f$) so we are adding up over the whole length of the wire. This form of the force will //always// work to find the magnetic force on the whole wire - we have not made very many assumptions so far in coming up with this equation. 
  
-However, if we do make a few assumptions we can simplify this equation significantly. We will start by //__assuming that the current in the wire is constant and in a steady state__//. This allows us to pull the current out of the integral, leaving:+However, if we do make a few assumptions we can simplify this equation significantly. We will start by //__assuming that the current in the wire is constant and in a steady state__//. This allows us to pull the current out of the integral as a constant, leaving:
 $$\vec{F}_{wire}= I \int_{l_i}^{l_f} d\vec{l} \times \vec{B}$$ $$\vec{F}_{wire}= I \int_{l_i}^{l_f} d\vec{l} \times \vec{B}$$
  
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 $$|\vec{F}_{wire}|=IBLsin(\theta)$$ $$|\vec{F}_{wire}|=IBLsin(\theta)$$
  
-{{  184_notes:week11_8.png?250}}+[{{  184_notes:week11_8.png?250|Use the right hand rule to determine the direction of the magnetic force F}}]
  
-where |$\vec{F}_{wire}$| is the magnitude of the force on the whole wire, I is the current through the wire, B is the //external// magnetic field, L is the length of the wire, and $\theta$ is the angle between the magnetic field and the length of the wire. //__Remember though that this equation is only good for straight wires with constant current that are in a constant magnetic field - because of all the assumptions that we made to get to this point.__//+where |$\vec{F}_{wire}$| is the magnitude of the force on the whole wire, $I$ is the current through the wire, $B$ is the //external// magnetic field, $L$ is the length of the wire, and $\theta$ is the angle between the magnetic field and the length of the wire. //__Remember though that this equation is only good for straight wires with constant current that are in a constant magnetic field - because of all the assumptions that we made to get to this point.__//
  
 To find the direction of the magnetic force, we will need to use the [[184_notes:rhr|right hand rule]]. Just like we did with the moving charge, you point your fingers in the direction of the moving charges (or rather in the direction of the current/$d\vec{l}$), then curl your fingers in the direction of the external magnetic field. Your thumb then points in the direction of the force. To find the direction of the magnetic force, we will need to use the [[184_notes:rhr|right hand rule]]. Just like we did with the moving charge, you point your fingers in the direction of the moving charges (or rather in the direction of the current/$d\vec{l}$), then curl your fingers in the direction of the external magnetic field. Your thumb then points in the direction of the force.
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 ==== Examples ==== ==== Examples ====
-[[:184_notes:examples:Week12_force_between_wires|Magnetic Force between Two Current-Carrying Wires]] +  * [[:184_notes:examples:Week12_force_between_wires|Magnetic Force between Two Current-Carrying Wires]] 
- +    * Video Example: Magnetic Force between Two Current-Carrying Wires 
-[[:184_notes:examples:Week12_force_loop_magnetic_field|Force on a Loop of Current in a Magnetic Field]]+  * [[:184_notes:examples:Week12_force_loop_magnetic_field|Force on a Loop of Current in a Magnetic Field]] 
 +    * Video Example: Force on a Loop of Current in a Magnetic Field 
 +{{youtube>BLuU7nZWv5k?large}} 
 +{{youtube>jMoOVHU6RR0?large}}
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