Chapter 29: Magnetic Fields Due to Currents

Figure 29-80 shows a cross-section of a long cylindrical conductor of radius $\mathit{a}\mathbf{=}\mathbf{4}\mathbf{ }\mathbf{\text{cm}}$containing a long cylindrical hole of radius$\mathit{b}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{50}\mathbf{}\mathbf{\text{cm}}$. The central axes of the cylinder and hole are parallel and are distance$\mathit{d}\mathbf{=}\mathbf{2}\mathbf{}\mathbf{\text{cm}}$apart; currentis uniformly distributed over the tinted area. (a) What is the magnitude of the magnetic field at the center of the hole? (b) Discuss the two special cases$\mathit{b}\mathbf{=}\mathbf{0}$and$\mathit{d}\mathbf{}\mathbf{=}\mathbf{}\mathbf{0}$.

The magnitude of the magnetic field $\mathbf{88}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{cm}$from the axis of a long straight wire is$\mathbf{7}\mathbf{.}\mathbf{30}\mathbf{}\mathbf{\mu T}$. What is the current in the wire?

Figure 29-81 shows a wire segment of length $\mathit{\Delta }\mathit{s}\mathbf{=}\mathbf{3}\mathbf{}\mathbf{\text{cm}}$, centered at the origin, carrying current $\mathit{i}\mathbf{=}\mathbf{2}\mathbf{}\mathbf{\text{A}}$ in the positive ydirection (as part of some complete circuit). To calculate the magnitude of the magnetic field produced by the segment at a point several meters from the origin, we can use $\overrightarrow{\mathbf{B}}\mathbf{=}\frac{{\mathbf{\mu }}_{\mathbf{0}}}{\mathbf{4}\mathbf{\pi }}\frac{\mathbf{i}\mathbf{\Delta }\overrightarrow{\mathbf{s}}\mathbf{\times }\hat{\mathbf{r}}}{{\mathbf{r}}^{\mathbf{2}}}$ as the Biot–Savart law. This is because r and u are essentially constant over the segment. Calculate (in unit-vector notation) at the$\left(x,y,z\right)$coordinates (a)localid="1663057128028" $(0,0,5m)$(b)localid="1663057196663" $(0,6m,0)$(c) localid="1663057223833" $\mathbf{(}\mathbf{7}\mathbf{}\mathbf{m}\mathbf{,}\mathbf{}\mathbf{7}\mathbf{m}\mathbf{,}\mathbf{}\mathbf{0}\mathbf{)}$and (d)$\left(-3\text{m},-4\text{m},0\right)$

In Fig. 29-83, two infinitely long wires carry equal currents i. Each follows a $\mathbf{90}\mathbf{°}$arc on the circumference of the same circle of radius R. Show that the magnetic field $\overrightarrow{\mathbf{B}}$at the center of the circle is the same as the field $\overrightarrow{\mathbf{B}}$a distance R below an infinite straight wire carrying a current Ito the left.

Figure 29-30 shows four circular Amperian loops (a, b, c, d) concentric with a wire whose current is directed out of the page. The current is uniform across the wire’s circular cross section (the shaded region). Rank the loops according to the magnitude of $\mathbf{\oint }\overrightarrow{\mathit{B}}\mathit{.}\overrightarrow{\mathit{d}\mathit{s}}$around each, greatest first.

A long wire is known to have a radius greater than$\mathbf{4}\mathbf{.}\mathbf{00}\mathbf{}\mathbf{}\mathbf{\text{mm}}$and to carry a current that is uniformly distributed over its cross section. The magnitude of the magnetic field due to that current is$\mathbf{0}\mathbf{.}\mathbf{28}\mathbf{}\mathbf{}\mathbf{\text{mT}}$at a point$\mathbf{4}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{\text{mm}}$from the axis of the wire, and$\mathbf{0}\mathbf{.}\mathbf{20}\mathbf{}\mathbf{mT}$at a point 10 mm from the axis of the wire. What is the radius of the wire?

Figure 29-84 shows a cross section of an infinite conducting sheet carrying a current per unit x-length of $\mathit{\lambda }$; the current emerges perpendicularly out of the page. (a) Use the Biot – Savart law and symmetry to show that for all pointsP above the sheet and all points $\mathit{P}\mathbf{\text{'}}$below it, the magnetic field$\overrightarrow{\mathbf{B}}$is parallel to the sheet and directed as shown. (b) Use Ampere’s law to prove that $\mathit{B}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{·}{\mathit{\mu }}_{\mathbf{0}}\mathit{\lambda }$ at all points P and$\mathit{P}\mathbf{\text{'}}$.

Figure 29-85 shows, in cross section, two long parallel wires that are separated by distance $\mathit{d}\mathbf{=}\mathbf{18}\mathbf{.}\mathbf{6}\mathbf{}\mathbf{\text{cm}}$. Each carries $\mathbf{4}\mathbf{.}\mathbf{23}\mathbf{}\mathbf{\text{A}}$, out of the page in wire 1 and into the page in wire 2. In unit-vector notation, what is the net magnetic field at point Pat distance $\mathit{R}\mathbf{=}\mathbf{34}\mathbf{.}\mathbf{2}\mathbf{}\mathbf{\text{cm}}$, due to the two currents?

In unit-vector notation, what is the magnetic field at pointPin Fig. 29-86 if$\mathit{i}\mathbf{=}\mathbf{10}\mathbf{ }\mathbf{\text{A}}$and$\mathit{a}\mathbf{}\mathbf{=}\mathbf{}\mathbf{8}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{ }\mathbf{\text{cm}}$? (Note that the wires are notlong.)

Three long wires all lie in an xyplane parallel to the xaxis. They are spaced equally,10 cm apart. The two outer wires each carry a current of 5.0 Ain the positive xdirection. What is the magnitude of the force on a3.0 m section of either of the outer wires if the current in the center wire is 3.2 A(a) in the positive xdirection and (b) in the negative xdirection?