In the figure shown, the conductor PQ of length \(l\) is moved from \(x = 0\) to \(x = b\) and then up to \(x = 2b\) with a constant velocity \(\vec{v}\). A uniform magnetic field \(\vec{B}\) is perpendicular to the plane of the paper and extends from \(x = 0\) to \(x = b\) and it is zero from \(x>b\). The magnitude of emf induced in the conductor is
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Motional emf only exists when a conductor "cuts" magnetic field lines. If the field is zero or if the conductor moves parallel to the field lines, the induced emf will be zero. Always check the boundaries of the magnetic field region.
Step 1: Understanding the Question:
The question asks for the magnitude of the induced emf in a rod moving with constant velocity through a region with a spatially limited magnetic field. Step 2: Key Formula or Approach:
The magnitude of motional emf (\( \varepsilon \)) induced in a conductor of length \( l \) moving with velocity \( v \) perpendicular to a uniform magnetic field \( B \) is:
\[ \varepsilon = Blv \] Step 3: Detailed Explanation:
1. Region \( 0 \le x \le b \): In this region, the conductor is moving through a uniform magnetic field \( B \). Since the velocity \( v \), field \( B \), and length \( l \) are mutually perpendicular, the induced motional emf is constant and is given by \( \varepsilon = Blv \).
2. Region \( b<x<2b \): According to the problem, the magnetic field \( B \) is zero in this region. Since \( B = 0 \), there is no magnetic flux change linked with the moving rod, and thus the induced emf is zero.
Looking at the options, option (3) correctly identifies the magnitude and the correct spatial interval where the field is present. Step 4: Final Answer:
The magnitude of emf is \( Blv \) for the interval \( 0 \le x \le b \).
