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.)

 The given figure, shows a long straight wire carrying current ${\mathrm{i}}_1=30.0 \mathrm{A}$ and a rectangular loop carrying current ${\mathrm{i}}_2=20.0 \mathrm{A}$. Take the dimensions to be $\mathrm{a}=1.00 \mathrm{cm}, \mathrm{b}=8.00 \mathrm{cm},$ and $\mathrm{L}=20.0 \mathrm{cm}.$ In unit-vector notation, what is the net force on the loop due to  ${\mathrm{i}}_1?$

Figure (a) shows, in cross-section, three current-carrying wires that are long, straight, and parallel to one another. Wires $1$ and $2$ are fixed in place on an $x$ axis, with separation $d$ . Wire 1 has a current of $0.750 \mathrm{A}$, but the direction of the current is not given. Wire $3,$ with a current of $0.250 \mathrm{A}$ out of the page, can be moved along the $x$ axis to the right of wire $2 .$ As wire$3$ is moved, the magnitude of the net magnetic force ${\overrightarrow{\mathrm{F}}}_2$ on wire $2$ due to the currents in wires $1$ and $3$ changes. The $x$ component of that force is $F_{2\mathrm{x}}$ and the value per unit length of wire $2$ $\frac{F_{2\mathrm{x}}}{L_2}$ is Figure (b)gives $\frac{F_{2\mathrm{x}}}{L_2}$ versus the position $x$ of wire  $3 .$ The plot has an asymptote$\frac{{\mathrm{F}}_{2\mathrm{x}}}{{\mathrm{L}}_2}=-0.627 \mu \mathrm{N} {\mathrm{m}}^{-1}$ as $x\to \infty .$ The horizontal scale is set by $x_{\mathrm{s}}=24.0 \mathrm{cm}.$ What are the
$\left(\mathrm{a}\right)$ size and
$\left(\mathrm{b}\right)$ direction (into or out of the page) of the current in wire $2 ?$

(a)                                                                                    (b)