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A uniform magnetic field B fills a cylindrical region of radius R. A straight metal rod CD of length l lies inside this region along a chord of the circular cross-section. If the magnitude of B increases at a constant rate dB/dt (direction unchanged), find the magnitude of the EMF induced in the rod CD.

1. (1/2)*(dB/dt)*l*sqrt(R² - l²/4)
2. (1/2)*(dB/dt)*l*R
3. (dB/dt)*l*sqrt(R² - l²/4)
4. (1/2)*(dB/dt)*l²

Correct answer: (1/2)*(dB/dt)*l*sqrt(R² - l²/4)

Solution

The time-varying field sets up an induced electric field E whose magnitude at radius r is E = (r/2)(dB/dt), directed tangentially (circular). For a straight rod, the line integral of E.dl reduces to (dB/dt) times the area of the triangle O-C-D, where O is the centre. That triangle has base l and height equal to the perpendicular distance d from the centre to the chord, with d = sqrt(R² - (l/2)²). Hence EMF = (dB/dt)*(1/2)*l*d.

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