91Ó°ÊÓ

Question: (a) Find the force on a square loop placed as shown in Fig. 5.24(a), near an infinite straight wire. Both the loop and the wire carry a steady current I.

(b) Find the force on the triangular loop in Fig. 5.24(b).

If B is uniform,show that $\mathit{A}\left(r\right)\mathbf{=}\mathbf{-}\frac{\mathbf{1}}{\mathbf{2}}\left(r×B\right)$works. That is, check that $\mathbf{∇}\mathbf{.}\mathit{A}\mathbf{=}\mathbf{0}$and$\mathbf{∇}\mathbf{×}\mathbf{A}\mathbf{=}\mathbf{B}$. Is this result unique, or are there other functions with the same divergence and curl?

(a) By whatever means you can think of (short of looking it up), find the vector potential a distance from an infinite straight wire carrying a current . Check that $\mathbf{∇}\mathbf{.}\mathbf{A}\mathbf{=}\mathbf{0}$and $\mathbf{∇}\mathbf{×}\mathbf{A}\mathbf{=}\mathbf{B}$.

(b) Find the magnetic potential inside the wire, if it has radius R and the current is uniformly distributed.

Find the vector potential above and below the plane surface current in Ex. 5.8.

Find and sketch the trajectory of the particle in Ex. 5.2, if it starts at

the origin with velocity

$$\begin{gathered} \mathbf{\left(}\mathbf{a}\mathbf{\right)}\stackrel{\mathbf{→}}{\mathbf{v}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\frac{\mathbf{E}}{\mathbf{B}}\stackrel{\mathbf{Áåœ}}{\mathbf{y}} \\ \mathbf{\left(}\mathbf{b}\mathbf{\right)}\stackrel{\mathbf{→}}{\mathbf{v}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\frac{\mathbf{E}}{\mathbf{2}\mathbf{B}}\stackrel{\mathbf{Áåœ}}{\mathbf{y}} \\ \mathbf{\left(}\mathbf{c}\mathbf{\right)}\stackrel{\mathbf{→}}{\mathbf{v}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\frac{\mathbf{E}}{\mathbf{B}}\mathbf{(}\stackrel{\mathbf{Áåœ}}{\mathbf{y}}\mathbf{+}\stackrel{\mathbf{Áåœ}}{\mathbf{z}}\mathbf{)}\mathbf{.} \end{gathered}$$

(a) Complete the proof of Theorem 2, Sect. 1.6.2. That is, show that any divergenceless vector field F can be written as the curl of a vector potential . What you have to do is find ${\mathbf{A}}_{\mathbf{x}\mathbf{}\mathbf{,}}{\mathbf{A}}_{\mathbf{y}}$and ${\mathit{A}}_{\mathbf{z}}$such that (i) $\mathbf{∂}{\mathbf{A}}_{\mathbf{z}}\mathbf{/}\mathbf{∂}\mathbf{y}\mathbf{-}\mathbf{∂}{\mathbf{A}}_{\mathbf{y}}\mathbf{/}\mathbf{∂}\mathbf{z}\mathbf{=}{\mathbf{F}}_{\mathbf{x}}$; (ii) $\mathbf{∂}{\mathbf{A}}_{\mathbf{x}}\mathbf{/}\mathbf{∂}\mathbf{z}\mathbf{-}\mathbf{∂}{\mathbf{A}}_{\mathbf{z}}\mathbf{/}\mathbf{∂}\mathbf{x}\mathbf{=}{\mathbf{F}}_{\mathbf{y}}$; and (iii) $\mathbf{∂}{\mathbf{A}}_{\mathbf{y}}\mathbf{/}\mathbf{∂}\mathbf{x}\mathbf{-}\mathbf{∂}{\mathbf{A}}_{\mathbf{x}}\mathbf{/}\mathbf{∂}\mathbf{y}\mathbf{=}{\mathbf{F}}_{\mathbf{z}}$. Here's one way to do it: Pick ${\mathit{A}}_{\mathbf{x}}\mathbf{=}\mathbf{0}$, and solve (ii) and (iii) for ${\mathit{A}}_{\mathbf{y}}$and ${\mathit{A}}_{\mathbf{z}}$. Note that the "constants of integration" are themselves functions of y and z -they're constant only with respect to x. Now plug these expressions into (i), and use the fact that $\mathbf{∇}\mathbf{â‹\dots }\mathit{F}\mathbf{=}\mathbf{0}\mathbf{}$to obtain

${\mathbf{A}}_{\mathbf{y}}\mathbf{=}{\mathbf{∫}}_{\mathbf{0}}^{\mathbf{x}}{\mathbf{F}}_{\mathbf{z}}\mathbf{(}{\mathbf{x}}^{\mathbf{\text{'}}}\mathbf{,}\mathbf{y}\mathbf{,}\mathbf{z}\mathbf{)}{\mathbf{dx}}^{\mathbf{\text{'}}}\mathbf{;}{\mathbf{A}}_{\mathbf{z}}\mathbf{=}{\mathbf{∫}}_{\mathbf{0}}^{\mathbf{y}}{\mathbf{F}}_{\mathbf{x}}\mathbf{(}\mathbf{0}\mathbf{,}{\mathbf{y}}^{\mathbf{\text{'}}}\mathbf{,}\mathbf{z}\mathbf{)}{\mathbf{dy}}^{\mathbf{\text{'}}}\mathbf{-}{\mathbf{∫}}_{\mathbf{0}}^{\mathbf{y}}{\mathbf{F}}_{\mathbf{y}}\mathbf{(}{\mathbf{x}}^{\mathbf{\text{'}}}\mathbf{,}\mathbf{y}\mathbf{,}\mathbf{z}\mathbf{)}{\mathbf{dx}}^{\mathbf{\text{'}}}$

(b) By direct differentiation, check that the you obtained in part (a) satisfies $\mathbf{∇}\mathbf{×}\mathbf{A}\mathbf{=}\mathbf{F}$. Is divergenceless? [This was a very asymmetrical construction, and it would be surprising if it were-although we know that there exists a vector whose curl is F and whose divergence is zero.]

(c) As an example, let $\mathbf{}\mathbf{F}\mathbf{=}\mathbf{y}\hat{\mathbf{x}}\mathbf{+}\mathbf{z}\hat{\mathbf{y}}\mathbf{+}\mathbf{x}\hat{\mathbf{z}}$. Calculate , and confirm that $\mathbf{∇}\mathbf{×}\mathbf{A}\mathbf{=}\mathbf{F}$. (For further discussion, see Prob. 5.53.)

(a) Check Eq. 5.76 for the configuration in Ex. 5.9.

(b) Check Eqs. 5.77 and 5.78 for the configuration in Ex. 5.11.

Prove Eq. 5.78, using Eqs. 5.63, 5.76, and 5.77. [Suggestion: I'd set up Cartesian coordinates at the surface, with Z perpendicular to the surface and X parallel to the current.]

Show that the magnetic field of a dipole can be written in coordinate-free form:

$\boxed{{\mathbf{B}}_{\mathbf{dip}}\mathbf{\left(}\mathbf{r}\mathbf{\right)}\mathbf{=}\frac{{\mathbf{μ}}_{\mathbf{0}}}{\mathbf{4}\mathbf{Ï€}}\frac{\mathbf{1}}{{\mathbf{r}}^{\mathbf{3}}}\mathbf{\left[}\mathbf{3}\mathbf{\right(}\mathbf{m}\mathbf{â‹\dots }\hat{\mathbf{r}}\mathbf{)}\hat{\mathbf{r}}\mathbf{-}\mathbf{m}\mathbf{]}}$

A circular loop of wire, with radius , R lies in the xy plane (centered at the origin) and carries a current running counterclockwise as viewed from the positive z axis.

(a) What is its magnetic dipole moment?

(b) What is the (approximate) magnetic field at points far from the origin?

(c) Show that, for points on the z axis, your answer is consistent with the exact field (Ex. 5.6), when $\mathbf{}\mathbf{z}\mathbf{>}\mathbf{>}\mathbf{R}$.