Changes between Version 2 and Version 3 of u/erica/d

Timestamp:

12/05/15 14:07:01 (9 years ago)

Author:

Erica Kaminski

Comment:

—

u/erica/d

@@ -16 +16 @@
 [[latex($\frac{1}{\mu}(\bold{B}\cdot \bold{\nabla}) \bold{B} = \hat{\bold{b}}\hat{\bold{b}} \cdot \bold{\nabla}(\frac{B^2}{2 \mu}) + \frac{B^2}{\mu} \frac{\hat{\bold{n}}}{R_c}$)]] 
 
-The first term on the RHS is a unit tensor dotted with the gradient of the magnetic pressure. The tensor is formed by the outer product of [[latex($\hat{\bold{b}}$)]] with itself. [[latex($\hat{\bold{b}}$)]] is the unit vector ''along'' the magnetic field line. 
+The first term on the RHS is a unit tensor dotted with the gradient of the magnetic pressure. The tensor is formed by the outer product of [[latex($\hat{\bold{b}}$)]] with itself. [[latex($\hat{\bold{b}}$)]] is the unit vector ''along'' the magnetic field line. Thus, if the field line was oriented along z, 
+
+[[latex($ \hat{\bold{b}}\hat{\bold{b}} = $)]]
+|| 0 || 0 || 0 || 
+||  0 ||0 || 0 ||
+|| 0 || 0 || 1 ||
+
+If the gradient of magnetic pressure was along the field line, that the first term on the LHS would be just the gradient of magnetic pressure. This would cancel with the other term for magnetic pressure. Thus we see that the magnetic pressure does not provide a force along field lines. It is thus does not provide an isotropic pressure force, but rather only acts perpendicular to the field lines, which is what we will consider next.
+
+Say the gradient of magnetic pressure is perpendicular to the magnetic field lines, say for example the gradient is along y (field is pointing along z still). Then,
+
+[[latex($\hat{\bold{b}}\hat{\bold{b}} \cdot \bold{\nabla}(\frac{B^2}{2 \mu}) = \bold{0}$)]]