A conducting rail is bent into the shape of a parabola y = kx^2 in the xy -plane. A horizontal conducting bar (parallel to the x -axis) moves upwards along the y -axis with a constant velocity v , starting from the origin at t=0 . A uniform magnetic field B exists perpendicular to the xy -plane. If the induced EMF in the loop is proportional to t^ 1 n , then the value of the integer n is ______.

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The bar moves along the y -axis with a constant velocity v . Its y -coordinate at any time t is given by y = vt . The rail is shaped as a parabola y = kx^2 . The active length of the bar is the distance between the two points where it intersects the parabola. For a given y , the x -coordinates of the intersection points are x = y k . The total active length of the bar is l = 2x = 2 y k = 2 vt k . The induced motional EMF in the bar is: E = B l v = B ( 2 vt k ) v = 2B v^3 k t^ 1 2 Thus, the induced EMF E t^ 1 2 . Co