4.1.11 Problems and solutions

The problems of this subsection are those of Lecture 8 (Problems 4.1.1–4.1.2, Home exercises 4.1.1–4.1.4, Worked examples 4.1.1–4.1.4) and of the exercise session of Lecture 9 (Problems 4.1.3–4.1.8, Home exercises 4.1.5–4.1.7), followed by twenty further exercises. Throughout, the force on a charge $q$ moving with velocity $v$ is the Lorentz force $F=q(E+v\times B)$, summed over the carriers of a straight wire of length $l$ it becomes $F=Il\times B$, and the field of a current distribution $J$ is given by Biot–Savart's law and its vector potential,

with $R$ pointing from the source point Q to the field point P. For a thin wire $Jd{v}^{\prime }\to Id{l}^{\prime }$. The two fundamental equations are $\nabla \times H=J$ (Ampère's formula, ${\oint }_{C}H\cdot dl=I_{\mathrm{enc}}$) and $\nabla \cdot B=0$, and the magnetic flux through a surface is $\Phi ={\int }_{S}B\cdot dS={\oint }_{C}A\cdot dl$. Every problem below starts by naming the symmetry. Constants: ${\mu }_0=4\pi \times 10^{-7}$ H/m (${\mu }_0/2\pi =2\times 10^{-7}$ H/m, ${\mu }_0/4\pi =10^{-7}$ H/m), $e=1.602\times 10^{-19}$ C, $m_e=9.109\times 10^{-31}$ kg, $m_p=1.6726\times 10^{-27}$ kg, and, as everywhere in this book, $1/(4\pi {\epsilon }_0)=9\times 10^9$ m/F, so that ${\mu }_0{\epsilon }_0=1/c^2$ with $c=3\times 10^8$ m/s. In the plots, the shaded bands mark the regions of the problem (conducting material in colour, empty space white).