Question

Show that the B-field outside of a long straight wire carrying a current I is derivable from the scalar potential $\varphi ^{*} = -(I/2\pi )\theta$ in cylindrical coordinates and that $\varphi ^{*}$ satisfies Laplace's equation. Why is this $\varphi ^{*}$ not one of the cylindrical harmonics (as would be the case for the electrostatic potential of a line charge)?

Please answer fully and show all steps. Thanks!

Answer 1

the gradient of f, namely

ög ög dg Ol

which mean

i i+-j or ^i+j+k

Suppose however, we are given f as a function of r and $theta_lc.gif$, that is, in polar coordinates, (or g in spherical coordinates, as a function of $phi_lc.gif$, $theta_lc.gif$, and $rho_lc.gif$).

For example, suppose

f-, or g , org sin θ

How do we find the gradient of f or g?

One way to find the gradient of such a function is to convert r or $rho_lc_bold.gif$ or $theta_lc_bold.gif$ into rectangular coordinates using the appropriate formulae for them, and perform the partial differentiation on the resulting expressions.

Thus we can write

-=(x2 +y2)-1/2

and find, by ordinary partial differentiating

$sections_eqn11.gif$

It is a bit more convenient sometimes, to be able to express the gradient directly in polar coordinates or spherical coordinates, like it is expressed in rectangular coordinates as above.

We want here an expression involving partial derivatives with respect to r and $theta_lc.gif$multiplied by vectors pointing respectively in the r direction, and $theta_lc_bold.gif$ direction.

So we want to know: what vectors should these partial derivatives be multiplied by in order to form the gradient?

When we find the answer, the actual partial derivative with respect to each polar variable will be the dot product of a unit vector in a polar direction with the gradient.

We therefore digress to discuss what thes unit vectors are so that you can recognize them.

The r direction is the direction tilted by an angle $theta_lc.gif$ counterclockwise from the x axis. A unit vector in that direction, call it u_r, can be written in any of the three following forms

$sections_eqn12.gif$

The unit vector in the $theta_lc.gif$ direction lies in the direction 90^o beyond the r direction, counterclockwisely, and is therefore given by

$sections_eqn13.gif$

1 0

The presence of a magnetic moment m creates a magnetic field which is the gradient of some scalar field. To gain a better intuitive feel about the relationship between scalar fields and their gradient vector fields, see Appendix A.3.6. Because the divergence of the magnetic field is zero, by definition, the divergence of the gradient of the scalar field is also zero, or ?²?_m = 0. The operator ?² is called the Laplacian and ?²?_m = 0 is Laplace