Topic: Faraday's law, induced emf and current· Electromagnetic Induction and Alternating Currents

A uniform magnetic field is restricted within a region of radius r. The magnetic field changes with time at a rate d B/dt. Loop 1 of radius R>r encloses the region r and loop 2 of radius R is outside the region of magnetic field as shown in the figur…

A uniform magnetic field is restricted within a region of radius \(r\). The magnetic field changes with time at a rate \(\frac{d\vec B}{dt}\). Loop 1 of radius \(R>r\) encloses the region \(r\) and loop 2 of radius \(R\) is outside the region of magnetic field as shown in the figure below. Then the e.m.f. generated is

two circles: loop 1 enclosing field region r, loop 2 outside

1. \(-\frac{d\vec B}{dt}\pi r^2\) in loop 1 and \(-\frac{d\vec B}{dt}\pi r^2\) in loop 2

2. \(-\frac{d\vec B}{dt}\pi R^2\) in loop 1 and zero in loop 2

3. \(-\frac{d\vec B}{dt}\pi r^2\) in loop 1 and zero in loop 2

4. zero in loop 1 and zero in loop 2

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Answer: 3. \(-\frac{d\vec B}{dt}\pi r^2\) in loop 1 and zero in loop 2

Induced emf is \(-\dfrac{d\Phi}{dt}\), and flux counts only the area where the field exists.

Loop 1 (radius \(R>r\)) encloses the whole field region, so \(\Phi=B\pi r^2\) and emf \(=-\dfrac{dB}{dt}\pi r^2\) (not \(\pi R^2\), since there is no field between \(r\) and \(R\)).
Loop 2 lies outside the field region, so its flux is zero and emf is zero.

Option (3).

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A uniform magnetic field is restricted within a region of radius r. The magnetic field changes with time at a rate d B/dt. Loop 1 of radius R>r encloses the region r and loop 2 of radius R is outside the region of magnetic field as shown in the figur…

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