Understanding the Direction of Magnetic Fields in Cartesian Coordinates

[nabliat]

here is the problem and solution:
http://i30.tinypic.com/2w2mnap.jpg

why they divide into two cases?

then they find a magnetic field

but we don't have any electricity flowing??

the i recognized the formula for a magnetic field from a wire
but when we find a the formula from the biotsavart law
the only direction is dlX\hat{r}

there is no \vec{\theta} there
because thus in Cartesian coordinates the direction always changes

how they get this theta direction thing??

 

[gabbagabbahey]

Remember, inductance is a purely geometrical quantity and is independent of the current flowing in the cable. This means you can apply any current you want, use it to calculate the flux and then divide by that current and the result will be the inductance. In that solution, they simply assume a constant current of I flowing up the inner cylinder and down the outer (uniformly distributed over the surface of each cylinder), which makes the calculation simple. You could just as well choose a time varying current or a non-uniformly distributed current if you like, but the calculation will be more difficult and the end result will be the same.

 

[nabliat]

how to get the theta direction??

 

[gabbagabbahey]

the theta direction points in the direction of increasing azimuthal angle, in cylindrical polar coordinates.

\mathbf{\hat{\theta}}=-\sin\theta\mathbf{\hat{x}}+\cos\theta\mathbf{\hat{y}}

this means that the field circles around the wire...there are many derivations of this in pretty much every EM text

 

[nabliat]

i got here a similar solution

http://i29.tinypic.com/hsuhde.jpg

but here they didnt specify whether the direction is with theta or
counter theta

i don't know how to diside that
with this new solution
??

 

[gabbagabbahey]

Use the 'right-hand rule'... point the thumb of your right hand in the direction of a straight current, and your fingers will curl in the direction of the field.

 

[nabliat]

thanks :)