Solved: PHYSICS Question for NEET

NEET PHYSICSMedium

A uniform magnetic field is restricted within a region of radius $r$ . The magnetic field changes with time at a rate $\frac{dB}{dt}$ . Loop 1 of radius $R$ ( $R > r$ ) encloses the region $r$ and Loop 2 of radius $R$ is entirely outside the region of the magnetic field as shown in the figure. Then the electromotive force (emf) generated is:

Zero in loop 1 and zero in loop 2

$-\frac{dB}{dt}\pi r^2$ in loop 1 and $-\frac{dB}{dt}\pi r^2$ in loop 2

$-\frac{dB}{dt}\pi r^2$ in loop 1 and zero in loop 2

$2\frac{dB}{dt}\pi r^2$ in loop 1 and zero in loop 2

Step-by-Step Solution

According to Faraday's Law of Induction, the induced emf is given by $\varepsilon = -\frac{d\Phi_B}{dt}$ .

Loop 1: This loop (Radius $R$ ) encloses the cylindrical region of the magnetic field (Radius $r$ ). Since the field $B$ exists only within radius $r$ , the total magnetic flux through Loop 1 is determined by the area of the magnetic field region, not the area of the loop itself. Flux $\Phi_1 = B \times (\text{Area of field}) = B(\pi r^2)$ . Induced emf $\varepsilon_1 = -\frac{d}{dt}(B\pi r^2) = -\pi r^2 \frac{dB}{dt}$ .
Loop 2: This loop is situated entirely outside the region of the magnetic field. Therefore, no magnetic field lines pass through it. Flux $\Phi_2 = 0$ . Induced emf $\varepsilon_2 = -\frac{d(0)}{dt} = 0$ .

Thus, a non-zero emf depends on the rate of change of field and the area of the field enclosed, while a loop enclosing no flux generates zero emf.

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