A small current-carrying coil, having a magnetic moment $\overrightarrow{{\mathit{p}}_{\mathit{m}}}$, is located at the axis of a round loop, of radius $R$, with current $I$ flowing through it. Find the magnitude of the vector force applied to the coil, if its distance from the centre of the loop is equal to $x$, and the vector $\overrightarrow{{\mathit{p}}_{\mathit{m}}}$ coincides in the direction with the axis of the loop.

A wire, bent as a parabola $y=a{x}^2$, is located in a uniform magnetic field of induction $\overrightarrow{B}$, the vector $\overrightarrow{\mathit{B}}$ being perpendicular to the plane $x-y$. At the moment $t=0$, a connector starts sliding translation-wise from the parabola apex, with a constant acceleration $\overrightarrow{w}$. Find the emf of electromagnetic induction in the loop thus formed as a function of $y$.

A rectangular loop with a sliding connector of length $I$, is located in a uniform magnetic field perpendicular to the loop plane. The magnetic induction is equal to $B$. The connector has an electric resistance $R$, the sides $AB$ and $CD$ have resistances $R_1$ and $R_2$, respectively. Neglecting the self-inductance of the loop, find the current flowing in the connector during its motion, with a constant velocity $v$.

A wire loop, enclosing a semi-circle of radius $a$, is located on the boundary of a uniform magnetic field of induction $\overrightarrow{B}$. At the moment $t=0$, the loop is set into rotation with a constant angular acceleration $\beta$, about an axis $O$, coinciding with a line of vector $\overrightarrow{\mathit{B}}$, on the boundary. Find the emf induced in the loop as a function of time $t$. Draw the approximate plot of this function. The arrow in the figure shows the emf direction, taken to be positive.

A long straight wire, carrying a current $I$, and a $\mathrm{U}$-shaped conductor, with a sliding connector, are located in the same plane, as shown in the figure. The connector, of length $L$ and resistance $R$, slides to the right with a constant velocity $v$. Find the current induced in the loop as a function of separation $r$ between the connector and the straight wire. The resistance of the $\mathrm{U}$-shaped conductor and the self inductance of the loop are assumed to be negligible.

A square frame, with side $a$, and a long straight wire, carrying a current $I$, are located in the same plane, as shown in the figure. The frame translates to the right with a constant velocity $U$. Find the emf induced in the frame as a function of distance $x$.