A region in the form of an equilateral triangle (in plane) of height has a uniform magnetic field pointing in the -direction. A conducting loop , in the form of an equilateral triangle of the same height , is placed in the plane with its vertex at in the orientation shown in the figure. At , the loop starts entering the region of the magnetic field with a uniform velocity along the -direction. The plane of the loop and its orientation remain unchanged throughout its motion.
Which of the following graph best depicts the variation of the induced emf () in the loop as a function of the distance () starting from ?
- A
- B
- C
- D
For , only a small triangular patch of the loop near vertex lies inside the field region. The overlap area grows quadratically as .
The flux is , so the induced emf is
The magnitude grows linearly with and the sign is negative (flux through the loop is increasing).
For , the loop and the field region overlap in a hexagonal star pattern. After subtracting the three corner triangles, the overlap area gives
This expression equals zero at which lies between and . The emf therefore starts negative, reaches its minimum at , rises linearly through zero at , and becomes positive, peaking near .
Graph (A) matches this behaviour.