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--- 184_notes:examples:week10_force_on_charge [2017/10/29 20:11] – created tallpaul
+++ 184_notes:examples:week10_force_on_charge [2017/11/02 13:32] (current) – dmcpadden
@@ Line 1: / Line 1: @@
-=====Magnetic Field from a Current Segment=====
+=====Magnetic Force on Moving Charge=====
-You may have read about how to find the [[184_notes:b_current#Magnetic_Field_from_a_Very_Long_Wire|magnetic field from a very long wire of current]]. Now, what is the magnetic field from a single segment? Suppose we have the configuration shown below. Your observation point is at the origin, and the segment of current $I$ runs in a straight line from $\langle -L, 0, 0 \rangle$ to $\langle 0, -L, 0 \rangle$.
+Suppose you have a moving charge ($q=1.5 \text{ mC}$) in a magnetic field ($\vec{B} = 0.4 \text{ mT } \hat{y}$). The charge has a speed of $10 \text{ m/s}$. What is the magnetic force on the charge if its motion is in the $+x$-direction? The $-y$-direction?
-{{ 184_notes:9_current_segment_bare.png?200 |Segment of Current}}
 ===Facts===
-  * The current in the segment is $I$.
+  * The charge is $q=1.5 \text{ mC}$.
-  * The observation point is at the origin.
+  * There is an external magnetic field $\vec{B} = 0.4 \text{ mT } \hat{y}$.
-  * The segment stretches from from $\langle -L, 0, 0 \rangle$ to $\langle 0, -L, 0 \rangle$.
+  * The velocity of the charge is $\vec{v} = 10 \text{ m/s } \hat{x}$ or $\vec{v} = -10 \text{ m/s } \hat{y}$.
 ===Lacking===
-  * $\vec{B}$
+  * $\vec{F}_B$
 ===Approximations & Assumptions===
-  * The current is steady, and the wire segment is uniform.
+  * The magnetic force on the charge contains no unknown contributions.
+  * The charge is moving at a constant speed (no other forces acting on it)
 ===Representations===
-  * We represent the Biot-Savart Law for magnetic field from a current as
+  * We represent the magnetic force on a moving charge as
-$$\vec{B}= \int \frac{\mu_0}{4 \pi}\frac{I \cdot d\vec{l}\times \vec{r}}{r^3}$$
+$$\vec{F}= q \vec{v} \times \vec{B}$$
-  * We represent the situation with diagram given above.
+  * We represent the two situations below.
+{{ 184_notes:10_moving_charge.png?500 |Moving Charge in a Magnetic Field}}
 ====Solution====
-Below, we show a diagram with a lot of pieces of the Biot-Savart Law unpacked. We show an example $\text{d}\vec{l}$, and a separation vector $\vec{r}$. Notice that $\text{d}\vec{l}$ is directed along the segment, in the same direction as the current. The separation vector $\vec{r}$ points as always from source to observation.
+Let's start with the first case, when $\vec{v}=10 \text{ m/s } \hat{x}$.
-{{ 184_notes:9_current_segment.png?400 |Segment of Current}}
+The trickiest part of finding magnetic force is the cross-product. You may remember from the [[184_notes:math_review#Vector_Multiplication|math review]] that there are a couple ways to do the cross product. Below, we show how to use vector components, for which it's helpful to rewrite $\vec{v}$ and $\vec{B}$ with their components.
-For now, we write $$\text{d}\vec{l} = \langle \text{d}x, \text{d}y, 0 \rangle$$
-and $$\vec{r} = \vec{r}_{obs} - \vec{r}_{source} = 0 - \langle x, y, 0 \rangle = \langle -x, -y, 0 \rangle$$
-Notice that we can rewrite $y$ as $y=-L-x$. This is a little tricky to arrive at, but is necessary to figure out unless you rotate your coordinate axes, which would be an alternative solution to this example. If finding $y$ is troublesome, it may be helpful to rotate. We can take the derivative of both sides to find $\text{d}y=-\text{d}x$. We can now plug in to express $\text{d}\vec{l}$ and $\vec{r}$ in terms of $x$ and $\text{d}x$:
-$$\text{d}\vec{l} = \langle \text{d}x, -\text{d}x, 0 \rangle$$
-$$\vec{r} = \langle -x, L+x, 0 \rangle$$
-Now, a couple other quantities that we see will be useful:
-$$\text{d}\vec{l} \times \vec{r} = \langle 0, 0, \text{d}x(L+x) - (-\text{d}x)(-x) \rangle = \langle 0, 0, L\text{d}x \rangle = L\text{d}x \hat{z}$$
-$$r^3 = (x^2 + (L+x)^2)^{3/2}$$
-The last thing we need is the bounds on our integral. Our variable of integration is $x$, since we chose to express everything in terms of $x$ and $\text{d}x$. Our segment begins at $x=-L$, and ends at $x=0$, so these will be the limits on our integral. Below, we write the integral all set up, and then we evaluate using some assistance some [[https://www.wolframalpha.com/input/?i=integral+from+-L+to+0+of+L%2F(x%5E2%2B(L%2Bx)%5E2)%5E(3%2F2)+dx|Wolfram Alpha]].
 \begin{align*}
-\vec{B} &= \int \frac{\mu_0}{4 \pi}\frac{I \cdot d\vec{l}\times \vec{r}}{r^3} \\
+  \vec{v} &= \langle 10, 0, 0 \rangle \text{ m/s} \\
-        &= \int_{-L}^0 \frac{\mu_0}{4 \pi}\frac{IL\text{d}x}{(x^2 + (L+x)^2)^{3/2}}\hat{z} \\
+  \vec{B} &= \langle 0, 4\cdot 10^{-4}, 0 \rangle \text{ T} \\
-        &= \frac{\mu_0}{2 \pi}\frac{I}{L}\hat{z}
+  \vec{v} \times \vec{B} &= \langle v_y B_z - v_z B_y, v_z B_x - v_x B_z, v_x B_y - v_y B_x \rangle \\
+                         &= \langle 0, 0, 4\cdot 10^{-3} \rangle \text{ T} \cdot \text{m/s}
 \end{align*}
+Alternatively, we could use the whole vectors and the angle between them. We find that we obtain the same result for the cross product. One would need to use the [[184_notes:rhr|Right Hand Rule]] to find that the direction of the cross product is $+\hat{z}$. The magnitude is given by
+$$\left|\vec{v} \times \vec{B} \right|= \left|\vec{v}\right| \left|\vec{B} \right| \sin\theta = (10 \text{ m/s})(4\cdot 10^{-4} \text{ T}) \sin 90^{\text{o}} = 4\cdot 10^{-3} \text{ T} \cdot \text{m/s}$$
+We get the same answer with both methods. Now, for the force calculation:
+$$\vec{F}_B = q \vec{v} \times \vec{B} = 1.5 \text{ mC } \cdot 4\cdot 10^{-3} \text{ T} \cdot \text{m/s } \hat{z} = 6 \mu\text{N}$$
+Notice that the $\sin 90^{\text{o}}$ is equal to $1$. This means that this is the maximum force that the charge can feel, and we get this value because the velocity and the B-field are exactly perpendicular. Any other orientation would have yielded a weaker magnetic force. In fact, in the second case, when the velocity is parallel to the magnetic field, the cross product evaluates to $0$. See below for the calculations.
+\begin{align*}
+  \vec{v} &= \langle 0, -10, 0 \rangle \text{ m/s} \\
+  \vec{B} &= \langle 0, 4\cdot 10^{-4}, 0 \rangle \text{ T} \\
+  \vec{v} \times \vec{B} &= \langle v_y B_z - v_z B_y, v_z B_x - v_x B_z, v_x B_y - v_y B_x \rangle \\
+                         &= \langle 0, 0, 0 \rangle
+\end{align*}
+Or, with whole vectors:
+$$\left|\vec{v} \times \vec{B} \right|= \left|\vec{v}\right| \left|\vec{B} \right| \sin\theta = (10 \text{ m/s})(4\cdot 10^{-4} \text{ T}) \sin 0 = 0$$
+When the velocity is parallel to the magnetic field, $\vec{F}_B=0$.