NEET Uncategorized MCQs & PYQs

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.

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 $l = 0.4 m$.
$e = 0.9 \times 7 \times 0.4 = 2.52 V$.

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 $I_0 = \frac{e_0}{R} = \frac{10^{-2} \times \pi(0.3)^2 \times 2\pi \times \frac{10}{3}}{\pi^2} = 10^{-2} \times 0.09 \times \frac{20}{3} = 6 \times 10^{-3} A = 6 mA$.

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.

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.

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.

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 $a < g$.

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 $I = 1 A$.
$U = \frac{1}{2} \times 0.1 \times 1^2 = 0.05 J$.

When a current flows through an inductor, a magnetic field is established around and within the coil.
The work done in establishing this current is stored in the magnetic field as magnetic potential energy ($U = \frac{1}{2} L I^2$).
Therefore, an inductor stores energy in its magnetic field.

Let current $I_1$ pass through the outer loop of radius $R_1$. The magnetic field at its centre is $B_1 = \frac{\mu_0 I_1}{2 R_1}$.
Since $R_1 >> R_2$, this field is nearly uniform over the inner loop of area $A_2 = \pi R_2^2$.
Flux linked is $\Phi_2 = B_1 A_2 = \frac{\mu_0 I_1 \pi R_2^2}{2 R_1} = M I_1$, hence $M \propto \frac{R_2^2}{R_1}$.