Question 31:
easy21. A metal ring is held horizontally and bar magnet is dropped through the ring with its length along the axis of the ring. The acceleration of the falling magnet is (1996)
As the magnet falls, magnetic flux linked with the ring increases.
By Lenz's law, an induced current is set up in the ring which opposes the downward motion of the magnet.
An upward repulsive magnetic force acts on the magnet, resulting in an acceleration $a < g$.
Question 32:
easy22. A big circular coil of 100 turns and average radius $10 m$ is rotating about its horizontal diameter at $2 rad s^{-1}$. If the vertical component of earth’s magnetic field at that place is $2 \times 10^{-5} T$ and electrical resistance of the coil is $12.56 \Omega$, then the maximum induced current in the coil will be : (2022)
The peak induced emf is given by $e_0 = N B A \omega = N B (\pi r^2) \omega$.
Substituting the values: $e_0 = 100 \times (2 \times 10^{-5}) \times (\pi \times 10^2) \times 2 = 4\pi \approx 12.56 V$.
The maximum induced current is $I_0 = \frac{e_0}{R} = \frac{12.56}{12.56} = 1 A$.
Question 33:
easy24. A metallic rod of mass per unit length $0.5 kg m^{-1}$ is lying horizontally on a smooth inclined plane which makes an angle of $30^\circ$ with the horizontal. The rod is not allowed to slide down by flowing a current through it when a magnetic field of induction $0.25 T$ is acting on it in the vertical direction. The current flowing in the rod to keep it stationary is (2018)
For the rod to remain stationary on the incline, the forces along the plane must balance: $m g \sin\theta = I l B \cos\theta$.
Rearranging gives $I = \left(\frac{m}{l}\right) \frac{g \tan\theta}{B}$.
With $\frac{m}{l} = 0.5 kg m^{-1}$, $g = 9.8 m s^{-2}$, $\theta = 30^\circ$, and $B = 0.25 T$: $I = \frac{0.5 \times 9.8 \times \tan(30^\circ)}{0.25} = \frac{4.9}{0.25 \sqrt{3}} \approx 11.32 A$.
Question 34:
easy27. A rectangular, a square, a circular and an elliptical loop, all in the (x-y) plane, are moving out of a uniform magnetic field with a constant velocity, $\vec{v} = v\hat{i}$. The magnetic field is directed along the negative z axis direction. The induced emf, during the passage of these loops, out of the field region, will not remain constant for: (2009)
The magnitude of induced emf is $e = B l_{eff} v$, where $l_{eff}$ is the length of the conductor cutting the field lines.
For rectangular and square loops, $l_{eff}$ is constant as they exit the field, resulting in a constant emf.
For circular and elliptical loops, $l_{eff}$ continuously varies with time, so the induced emf does not remain constant.
Question 35:
easy28. A straight line conductor of length $0.4 m$ is moved with a speed of $7 m/s$ perpendicular to a magnetic field of intensity $0.9 Wb/m^2$. The induced e.m.f. across the conductor is (1995)
The motional e.m.f. induced across a straight conductor moving perpendicular to a magnetic field is $e = B v l$.
Given $B = 0.9 Wb/m^2$, $v = 7 m/s$, and $l = 0.4 m$.
$e = 0.9 \times 7 \times 0.4 = 2.52 V$.
Question 36:
easy29. In a region of magnetic induction $B = 10^{-2} tesla$, a circular coil of radius $30 cm$ and resistance $\pi^2 ohm$ is rotated about an axis which is perpendicular to the direction of B and which forms a diameter of the coil. If the coil rotates at $200 rpm$ the amplitude of the alternating current induced in the coil is (1988)
The maximum induced emf is $e_0 = B A \omega = B (\pi r^2) (2\pi f)$.
Here $B = 10^{-2} T$, $r = 0.3 m$, $f = \frac{200}{60} = \frac{10}{3} s^{-1}$, and $R = \pi^2 \Omega$.
Amplitude of induced current is $I_0 = \frac{e_0}{R} = \frac{10^{-2} \times \pi(0.3)^2 \times 2\pi \times \frac{10}{3}}{\pi^2} = 10^{-2} \times 0.09 \times \frac{20}{3} = 6 \times 10^{-3} A = 6 mA$.
Question 37:
easy30. In which of the following devices, the eddy current effect is not used? (2019)
Eddy currents are utilized in induction furnaces (for melting metals), magnetic braking in trains, and dead-beat galvanometers.
An electric heater works purely on the principle of Joule heating due to current flowing through a high-resistance wire element.
Therefore, the eddy current effect is not used in an electric heater.
Question 38:
easy31. Eddy currents, are produced when (1988)
Eddy currents are circulating currents induced in bulk pieces of conductors (metals).
According to Faraday's law of induction, these currents are set up when there is a change in the magnetic flux linked with the metal.
Hence, they are produced when a bulk metal is placed in a time-varying magnetic field.
Question 39:
easy20. A wire loop is rotated in a magnetic field. The frequency of change of direction of the induced e.m.f. is: (2013)
The induced emf in a rotating loop varies sinusoidally as $e = e_0 \sin(\omega t)$.
In one complete revolution ($2\pi$ radians), the sine function alternates between positive and negative half-cycles.
Thus, the direction of the induced e.m.f. reverses twice per revolution.
Question 40:
easy40. A 100 millihenry coil carries a current of $1 A$. Energy stored in its magnetic field is (1991)
Energy stored in the magnetic field of an inductor is $U = \frac{1}{2} L I^2$.
Here, $L = 100 mH = 0.1 H$ and $I = 1 A$.
$U = \frac{1}{2} \times 0.1 \times 1^2 = 0.05 J$.