A metallic rod of 1 m length is rotated with a frequency of 50 rev/s, with one end hinged at the centre and the other end at the circumference of a circular metallic ring of radius 1 m, about an axis passing through the centre and perpendicular to the plane of the ring (as shown in the figure). A constant and uniform magnetic field of 1 T parallel to the axis is present everywhere. What is the emf between the centre and the metallic ring?

1. 150 V

2. 130 V

3. 157 V

4. 133V

A wheel with \(10\) metallic spokes each \(0.5\) m long is rotated with a speed of \(120\) rev/min in a plane normal to the horizontal component of earth’s magnetic field H_E at a place. If \(H_E=0.4\) G at the place, what is the induced emf between the axle and the rim of the wheel? (\(1\) G=\(10^{-4}\) T)
1. \(5.12\times10^{-5}\) T
2. \(0\)
3. \(3.33\times10^{-5}\)
4. \(6.28\times10^{-5}\)

Refer to figure. The arm PQ of the rectangular conductor is moved from x = 0, outwards. The uniform magnetic field is perpendicular to the plane and extends from x = 0 to x = b and is zero for x > b. Only the arm PQ possesses substantial resistance r. Consider the situation when the arm PQ is pulled outwards from x = 0 to x = 2b with a constant speed v. The induced emf is:

$1. -\mathrm{Blv} \mathrm{for} 0\le \mathrm{x}<\mathrm{b}, 0 \mathrm{for} \mathrm{b}\le \mathrm{x}<2\mathrm{b}$
$2. +\mathrm{Blv} \mathrm{for} 0\le \mathrm{x}<\mathrm{b}, 0 \mathrm{for} \mathrm{b}\le \mathrm{x}<2\mathrm{b}$
$3. -\mathrm{Blv} \mathrm{for} \mathrm{b}\le \mathrm{x}<2\mathrm{b}, 0 \mathrm{for} 0\le \mathrm{x}<\mathrm{b}$
$4. +\mathrm{Blv} \mathrm{for} \mathrm{b}\le \mathrm{x}<2\mathrm{b}, 0 \mathrm{for} 0\le \mathrm{x}<\mathrm{b}$

Refer to the figure. The arm PQ of the rectangular conductor is moved from x = 0, outwards. The uniform magnetic field is perpendicular to the plane and extends from x = 0 to x = b and is zero for x > b. Only the arm PQ possesses substantial resistance r. Consider the situation when the arm PQ is pulled outwards from x = 0 to x = 2b with constant speed v. The force necessary to pull the arm is:

$1. \frac{{\mathrm{B}}^2{\mathrm{l}}^2\mathrm{v}}{\mathrm{r}} \mathrm{for} 0\le \mathrm{x}<\mathrm{b}, 0 \mathrm{for} \mathrm{b}\le \mathrm{x}<2\mathrm{b}$
$2. \frac{{\mathrm{B}}^2{\mathrm{l}}^2\mathrm{v}}{2\mathrm{r}} \mathrm{for} 0\le \mathrm{x}<\mathrm{b}, 0 \mathrm{for} \mathrm{b}\le \mathrm{x}<2\mathrm{b}$
$3. 0 \mathrm{for} 0\le \mathrm{x}<\mathrm{b}, \frac{{\mathrm{B}}^2{\mathrm{l}}^2\mathrm{v}}{\mathrm{r}} \mathrm{for} \mathrm{b}\le \mathrm{x}<2\mathrm{b}$
$4. 0 \mathrm{for} 0\le \mathrm{x}<\mathrm{b}, \frac{{\mathrm{B}}^2{\mathrm{l}}^2\mathrm{v}}{2\mathrm{r}} \mathrm{for} \mathrm{b}\le \mathrm{x}<2\mathrm{b}$

Refer to figure. The arm PQ of the rectangular conductor is moved from x = 0, outwards. The uniform magnetic field is perpendicular to the plane and extends from x = 0 to x = b and is zero for x > b. Only the arm PQ possesses substantial resistance r. Consider the situation when the arm PQ is pulled outwards from x = 0 to x = 2b and is then moved back to x = 0 with constant speed v. The power dissipated as Joule heat is: 

$1. \frac{{\mathrm{B}}^2{\mathrm{l}}^2\mathrm{v}}{2\mathrm{r}} \mathrm{for} 0\le \mathrm{x}<\mathrm{b}, 0 \mathrm{for} \mathrm{b}\le \mathrm{x}<2\mathrm{b}$
$2. \frac{{\mathrm{B}}^2{\mathrm{l}}^2{\mathrm{v}}^2}{\mathrm{r}} \mathrm{for} 0\le \mathrm{x}<\mathrm{b}, 0 \mathrm{for} \mathrm{b}\le \mathrm{x}<2\mathrm{b}$
$3. 0 \mathrm{for} \mathrm{b}\le \mathrm{x}<2\mathrm{b}, \frac{{\mathrm{B}}^2{\mathrm{l}}^2{\mathrm{v}}^2}{\mathrm{r}} \mathrm{for} 0\le \mathrm{x}<\mathrm{b}$
$4. 0 \mathrm{for} \mathrm{b}\le \mathrm{x}<2\mathrm{b}, \frac{{\mathrm{B}}^2{\mathrm{l}}^2{\mathrm{v}}^2}{2\mathrm{r}} \mathrm{for} 0\le \mathrm{x}<\mathrm{b}$

Two concentric circular coils, one of small radius \({r_1}\) and the other of large radius \({r_2},\) such that \({r_1<<r_2},\)  are placed co-axially with centres coinciding. The mutual inductance of the arrangement is:
1. \(\frac{\mu_0\pi r_1^2}{3r_2}\)
2. \(\frac{2\mu_0\pi r_1^2}{r_2}\)
3. \(\frac{\mu_0\pi r_1^2}{r_2}\)
4. \(\frac{\mu_0\pi r_1^2}{2r_2}\)

The expression for the magnetic energy stored in a solenoid in terms of magnetic field \(B\), area \(A\) and length \(l\) of the solenoid is:
1. \( \frac{1}{\mu_0}B^2Al\)
2. \( \frac{1}{2\mu_0}B^2Al\)
3. \( \frac{2}{\mu_0}B^2Al\)
4. \( \frac{3}{2\mu_0}B^2Al\)

The ratio of magnetic energy per unit volume and electrostatic energy stored per unit volume in a parallel plate capacitor is:

$1. \frac{1}{{\mathrm{\epsilon }}_0{\mathrm{\mu }}_0}{\left(\frac{\mathrm{B}}{\mathrm{E}}\right)}^2$
$2. \frac{1}{{\mathrm{\epsilon }}_0{\mathrm{\mu }}_0}{\left(\frac{\mathrm{E}}{\mathrm{B}}\right)}^2$
$3. \frac{1}{2{\mathrm{\epsilon }}_0{\mathrm{\mu }}_0}{\left(\frac{\mathrm{B}}{\mathrm{E}}\right)}^2$
$4. \frac{1}{2{\mathrm{\epsilon }}_0{\mathrm{\mu }}_0}{\left(\frac{\mathrm{E}}{\mathrm{B}}\right)}^2$

Kamla peddles a stationary bicycle. The pedals of the bicycle are attached to a \(100\) turn coil of an area of \(0.10~\text{m}^2\). The coil rotates at half a revolution per second and it is placed in a uniform magnetic field of \(0.01~\text{T}\) perpendicular to the axis of rotation of the coil. What is the maximum voltage generated in the coil?
1. \(0.628~\text{V}\)
2. \(0.421~\text{V}\)
3. \(0.314~\text{V}\)
4. \(0\) 

The polarity of the capacitor in the given figure is: