Question

A conducting ring of circular cross-section with inner and outer radii $a$ and $b$ is made out of a material of resistivity $\rho$ . The thickness of the ring is $h$ . It is placed coaxially in a vertical cylindrical region of magnetic field, $B=krt$ , where $k$ is a positive constant, $r$ is the distance from the axis and $t$ is the time. If the current through the ring is, $I=\left(\frac{kh}{\alpha p}\right)\left[b^3-a^3\right]$ , then what is the value of $\alpha$ ?

Answer 1

Answer: 9

The magnetic flux through the area of radius $r$ is,
$\varphi ={\int }_0^{r}krt2\pi rdr=\frac{2\pi kt{r}^3}{3}$ .
Emf induced is the rate of change of the magnetic flux,
$e=\left|\frac{d\varphi }{dt}\right|=\frac{2\pi k{r}^3}{3}$ .
The net induced current in terms of emf and resistance,
$I={\int }_a^b\frac{2\pi r^{3}khdr}{3(2\pi r)}=\frac{kh}{9\rho }\left(b^3-a^3\right)$ .