184_notes:b_shapes

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184_notes:b_shapes [2021/06/16 19:21] – [Solenoid] bartonmo184_notes:b_shapes [2021/06/16 19:23] – [Coils] bartonmo
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 [{{184_notes:10_ring_dl_notes.png?250|$dl$ in a coil  }}] [{{184_notes:10_ring_dl_notes.png?250|$dl$ in a coil  }}]
  
-In a circular coil, the current through the wire would run either clockwise or counter-clockwise around the loop. If we wanted to calculate the magnetic field from that current, we could use the Biot-Savart Law+In a circular coil, the current through the wire would run either **clockwise or counter-clockwise around the loop**. If we wanted to calculate the magnetic field from that current, we could use the Biot-Savart Law
 $$\vec{B}_{tot}= \int \frac{\mu_0}{4 \pi}\frac{I \cdot d\vec{l}\times \hat{r}}{r^2}$$ $$\vec{B}_{tot}= \int \frac{\mu_0}{4 \pi}\frac{I \cdot d\vec{l}\times \hat{r}}{r^2}$$
 where the $d\vec{l}$ now would need to represent a small piece of the circular loop, rather than a small piece of the straight wire. This is shown in the figure above, where the circular coil now is drawn in the xy-plane and the z-direction points straight through the center of the coil.  where the $d\vec{l}$ now would need to represent a small piece of the circular loop, rather than a small piece of the straight wire. This is shown in the figure above, where the circular coil now is drawn in the xy-plane and the z-direction points straight through the center of the coil. 
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 [[184_notes:examples:week10_current_ring|The example below]] shows the calculation for the magnetic field at the center of a circular coil for a counterclockwise current. Because the circular coil is highly symmetric, we get a fairly simple answer for the magnetic field at the center of the coil (in the plane of the coil): [[184_notes:examples:week10_current_ring|The example below]] shows the calculation for the magnetic field at the center of a circular coil for a counterclockwise current. Because the circular coil is highly symmetric, we get a fairly simple answer for the magnetic field at the center of the coil (in the plane of the coil):
 $$\vec{B}=\frac{\mu_0 I}{2R} \hat{z}$$ $$\vec{B}=\frac{\mu_0 I}{2R} \hat{z}$$
-Which depends only on the current going through the coil ($I$) and the radius of the coil ($R$). The direction of the magnetic field comes from the cross product of the $\vec{r}$ with each $d\vec{l}$ in the coil, which we can check with the right hand rule. If you point your fingers in the direction of the current at any point on the coil and curl them toward the center of the circular coil, your thumb will always be pointing up in the $+\hat{z}$ direction. //__Note this equation is only true for the magnetic field at the center of a circular coil__// - if you have a different observation point or a different shape of coil, this equation will no longer give you an accurate magnetic field. +Which depends only on the current going through the coil $I$ and the radius of the coil $R$. The direction of the magnetic field comes from the cross product of the $\vec{r}$ with each $d\vec{l}$ in the coil, which we can check with the right hand rule. If you point your fingers in the direction of the current at any point on the coil and curl them toward the center of the circular coil, your thumb will always be pointing up in the $+\hat{z}$ direction. //__Note this equation is only true for the magnetic field at the center of a circular coil__// - if you have a different observation point or a different shape of coil, this equation will no longer give you an accurate magnetic field. 
  
 [{{  184_notes:10_ring_with_b_z.png?200|Magnetic field at a point above the coil}}] [{{  184_notes:10_ring_with_b_z.png?200|Magnetic field at a point above the coil}}]
  • 184_notes/b_shapes.txt
  • Last modified: 2021/06/16 19:24
  • by bartonmo