From ebdf24e3a116e475bc4e2b5b82627169d8b4bd83 Mon Sep 17 00:00:00 2001
From: Erik Strand <erik.strand@cba.mit.edu>
Date: Wed, 6 Mar 2019 15:01:40 -0500
Subject: [PATCH] Answer 6.4

---
 _psets/4.md | 33 ++++++++++++++++++++++++++++++++-
 1 file changed, 32 insertions(+), 1 deletion(-)

diff --git a/_psets/4.md b/_psets/4.md
index 3103c6b..09d314b 100644
--- a/_psets/4.md
+++ b/_psets/4.md
@@ -189,18 +189,49 @@ Yes, there is a sign error that I'm ignoring for now.
 The ampere was formerly defined [BIPM, 2014] as "The ampere is that constant current which, if
 maintained in two straight parallel conductors of infinite length, of negligible circular
 cross-section, and placed 1 metre apart in vacuum, would produce between these conductors a force
-equal to 2 × 10−7 newton per metre of length."
+equal to $$\num{2e-7}$$ newton per metre of length."
 
 ### (a)
 
 {:.question}
 Show that that current at that distance produces that force.
 
+First let's find the magnetic field of an infinitely long straight conductor. Considering the
+Biot-Savart Law and the symmetry of this problem, the magnetic field must have no component parallel
+to the wire, must always be perpendicular to the radial separation between the test point and the
+wire, and must have a constant magnitude at each radius. Consider then a circle of radius $$r$$
+centered on the wire. Amp&egrave;re's Law tells us that the magnitude of the field on this circle is
+$$I / (2 \pi r)$$.
+
+The differential force exerted by this field on a differential piece of current is $$dF = I (dl
+\times B)$$. In this case the direction of the current and the magnetic field are perpendicular, so
+the direction of the cross product is always toward the other wire. The magnitude of the force per
+meter is
+
+$$
+\begin{align*}
+F &= \frac{\mu_0 I^2}{2 \pi r} \\
+&= \frac{4 \pi \times 10^{-7} \si{H/m} \cdot 1 \si{A^2}}{2 \pi \cdot 1 \si{m}} \\
+&= \num{2e-7} \si{N}
+\end{align*}
+$$
+
+Note that we don't need to multiply this by two to account for the force wire two exerts on wire
+one. Newton's [third
+law](https://en.wikipedia.org/wiki/Newton%27s_laws_of_motion#Newton's_third_law) guarantees that
+this is equal and opposite, so it only makes sense to talk about the force exerted by one on the
+other. Maxwell's equations respect this (otherwise it would be easy to build a perpetual motion
+machine).
+
 ### (b)
 
 {:.question}
 What is the problem with defining the ampere this way?
 
+It's not a very practical experiment. No wires are infinitely long, so you'll necessarily have
+fringe fields to factor into your results. No conductors are infinitely thin. You have to hold the
+wires and supply current to them somehow, so they can't be totally surrounded by a vacuum.
+
 
 ## (6.5)
 
-- 
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