In the figure below, five long parallel wires in $xy$ and plane are separated by distance $d=50.0 \mathrm{cm}$. The currents into the page are $i_1=2.00 \mathrm{A},$ $i_3=0.250 \mathrm{A}, i_4=6.00 \mathrm{A},$ and $i_5=2.00 \mathrm{A};$ the current out of the page is $i_2=4.00 \mathrm{A}$. What is the magnitude of the net force per unit length acting on wire $3$ due to the currents in the other wires?

 Figure (a) shows, in cross section, two long, parallel wires carrying current and separated by distance $L .$ The ratio$\frac{i_1}{i_2}$ of their currents is $4.00 ;$ the directions of the currents are not indicated. Figure (b) shows the $\mathrm{y}$ component $B_{\mathrm{y}}$ of their net magnetic field along the $x$ axis to the right of wire $2 .$ The vertical scale is set by$B_{\mathrm{ys}}=4.0 \mathrm{nT},$ and the horizontal scale is set by $x_{\mathrm{s}}=40.0 \mathrm{cm}$
(a) At what value of $x>0$ is $B_{\mathrm{y}}$ maximum?

(b) If ${\mathrm{i}}_2=3 \mathrm{mA},$ what is the value of that maximum? What is the direction (into or out of the page) of (c) $i_1$ and (d) $i_2?$

(a)                                                              (b)

In figure, a conductor carries $2.0 \mathrm{A}$ along the closed path $abcdefgha$, running along $8$ of the $12$ edges of a cube of edge length $10 \mathrm{cm}$.
Taking the path to be a combination of three square current loops $(bcfgb,$ $abgha$ and $cdefc$), find the net magnetic moment of the path in unit-vector notation.

The figure shows two closed paths wrapped around two conducting loops carrying current $i_1=6.0 \mathrm{A}$ and $i_2=3.0 \mathrm{A}$ respectively. What is the value of the integral$\oint \overrightarrow{B}·d\overrightarrow{s}$ for$\left(a\right)$ path $1$ and$\left(b\right)$ path $2$?

The figure shows wire $1$ in cross section. The wire is long and straight, carries a current of $2.50 \mathrm{mA}$ out of the page and is at a distance, $d_1=4.00 \mathrm{cm}$ from the surface. Wire $2$, which is parallel to wire $1$ and also long, is at horizontal distance, $d_2=5.00 \mathrm{cm}$ from wire $1$ and rise a current $6.80 \mathrm{mA}$ into the page. What is the $x$ component of the magnetic force per unit length on wire $2$ due to wire $1$?

A toroid having a square cross-section, $5.00 \mathrm{cm}$ on a side, and an inner radius of $19.0 \mathrm{cm}$ has $460$ turns and carries a current of $0.400 \mathrm{A}$ . (It is made up of a square solenoid-instead of round one as in figure bent into a doughnut shape). What is the magnetic field inside the toroid at (a) the inner radius and

(b) the outer radius?

The given figure shows an arrangement known as Helmholtz coil. It consists of two circular coaxial coils, each of $200$ turns and radius $R=20.0 \mathrm{cm},$ separated by a distance $S=R .$ The two coils carry equal currents $i=20.2 \mathrm{mA}$ in the same direction. Find the magnitude of the net magnetic field at $P$, midway between the coils.

The given figure shows a cross-section of a long thin ribbon of width $\mathrm{w}=6.20 \mathrm{cm}$ that is carrying a uniformly distributed current $\mathrm{i}=4.61 \mathrm{\mu A}$ into the page. In unit-vector notation, what is the magnetic field $\overrightarrow{\mathrm{B}}$ at a point $\mathrm{P}$ in the plane of the ribbon at a distance $\mathrm{d}=1.61 \mathrm{cm}$ from its edge? (Hint: Imagine the ribbon as being constructed from many long, thin, parallel wires.)