184_notes:moving_q

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184_notes:moving_q [2018/05/15 16:04] – curdemma184_notes:moving_q [2018/07/03 03:49] – [Magnetic Field Equation for a Moving Point Charge] curdemma
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 ==== Magnetic Field Equation for a Moving Point Charge ==== ==== Magnetic Field Equation for a Moving Point Charge ====
  
-The general equation for the magnetic field ($\vec{B}$) at some Point P due to a moving charge is given by:+The general equation for the magnetic field ($\vec{B}$), with units of Tesla ($T$), at some Point $P$ due to a moving charge is given by:
 $$\vec{B}=\frac{\mu_0}{4 \pi}\frac{q\vec{v}\times \hat{r}}{r^2}$$ $$\vec{B}=\frac{\mu_0}{4 \pi}\frac{q\vec{v}\times \hat{r}}{r^2}$$
 which you may hear referred to as the [[https://en.wikipedia.org/wiki/Biot%E2%80%93Savart_law|Biot-Savart Law]] in other resources or textbooks. This equation has several pieces to it: which you may hear referred to as the [[https://en.wikipedia.org/wiki/Biot%E2%80%93Savart_law|Biot-Savart Law]] in other resources or textbooks. This equation has several pieces to it:
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   * **Separation Distance** - the $r$ in this equation should feel very familiar from all the work we did with electric fields and potentials. In this equation, this is still the same magnitude of the separation vector that points from the source (the moving charge in this case) to the observation point (P in this case). Mathematically, we can still find the magnitude of this vector using:   * **Separation Distance** - the $r$ in this equation should feel very familiar from all the work we did with electric fields and potentials. In this equation, this is still the same magnitude of the separation vector that points from the source (the moving charge in this case) to the observation point (P in this case). Mathematically, we can still find the magnitude of this vector using:
 $$r=|\vec{r}|=|\vec{r_p}-\vec{r_q}|$$ $$r=|\vec{r}|=|\vec{r_p}-\vec{r_q}|$$
-{{  184_notes:Week9_separationvector.png?300}}+[{{  184_notes:Week9_separationvector.png?300|Separation Vector}}]
   * **Separation Unit Vector** - Similarly the $\hat{r}$ in this equation is the unit vector (has a magnitude of one) that points in the same direction as $\vec{r}$. From before, you know that you can calculate the unit vector using the separation vector and it's magnitude: $$\hat{r}=\frac{\vec{r}}{r}$$ We could also use this unit vector definition to rewrite the magnetic field equation so it is in terms of the $\vec{r}$ rather than $\hat{r}$. $$\vec{B}=\frac{\mu_0}{4 \pi}\frac{q\vec{v}\times \hat{r}}{r^2}=\frac{\mu_0}{4 \pi}\frac{q\vec{v}\times \vec{r}}{r^3}$$   * **Separation Unit Vector** - Similarly the $\hat{r}$ in this equation is the unit vector (has a magnitude of one) that points in the same direction as $\vec{r}$. From before, you know that you can calculate the unit vector using the separation vector and it's magnitude: $$\hat{r}=\frac{\vec{r}}{r}$$ We could also use this unit vector definition to rewrite the magnetic field equation so it is in terms of the $\vec{r}$ rather than $\hat{r}$. $$\vec{B}=\frac{\mu_0}{4 \pi}\frac{q\vec{v}\times \hat{r}}{r^2}=\frac{\mu_0}{4 \pi}\frac{q\vec{v}\times \vec{r}}{r^3}$$
-  * **Velocity** - unlike what we did before, the magnetic field equation depends on the velocity of the moving charge. Remember that //velocity is vector//, so it has both a magnitude and direction. The units of velocity here are m/s like normal. Here we have //__assumed that velocity is constant__// or //__assumed that we are looking at a particular instant in time__//, so there is a single, constant vector to use for $\vec{v}$.+  * **Velocity** - unlike what we did before, the magnetic field equation depends on the velocity of the moving charge. Remember that //velocity is vector//, so it has both a magnitude and direction. The units of velocity here are m/s like normal. Here we have //__assumed that velocity is constant__// or //__assumed that we are looking at a particular instant in time__//, so there is a single, constant vector to use for $\vec{v}$ (units of $\frac{m}{s}$).
   * **Direction - Cross Product** - the final piece to this equation is the [[183_notes:cross_product|cross product]] between $\vec{v}$ and $\hat{r}$ or between $\vec{v}$ and $\vec{r}$ depending on how you wrote the equation. The cross product is a way to [[184_notes:math_review#vector_multiplication|multiply vectors]] that gives a perpendicular vector as the product. You have seen the cross product before when we were talking about [[184_notes:e_flux|area vectors]] or you may remember the cross product from calculus or learning [[183_notes:torque|torque]] in mechanics. Because of how the cross product works, this means that the **magnetic field is perpendicular to both the velocity of the moving charge and the separation vector**. So for a charge moving in a straight line, the magnetic field creates a curling field (in ring) around the charge. You can always use the [[183_notes:cross_product|determinant method]] to calculate the cross product and thus the direction of the magnetic field, but we will also make use of the Right Hand Rule as tool to figure out the direction of the magnetic field.   * **Direction - Cross Product** - the final piece to this equation is the [[183_notes:cross_product|cross product]] between $\vec{v}$ and $\hat{r}$ or between $\vec{v}$ and $\vec{r}$ depending on how you wrote the equation. The cross product is a way to [[184_notes:math_review#vector_multiplication|multiply vectors]] that gives a perpendicular vector as the product. You have seen the cross product before when we were talking about [[184_notes:e_flux|area vectors]] or you may remember the cross product from calculus or learning [[183_notes:torque|torque]] in mechanics. Because of how the cross product works, this means that the **magnetic field is perpendicular to both the velocity of the moving charge and the separation vector**. So for a charge moving in a straight line, the magnetic field creates a curling field (in ring) around the charge. You can always use the [[183_notes:cross_product|determinant method]] to calculate the cross product and thus the direction of the magnetic field, but we will also make use of the Right Hand Rule as tool to figure out the direction of the magnetic field.
  
  • 184_notes/moving_q.txt
  • Last modified: 2021/07/05 21:51
  • by schram45