A rod of length 60 cm rotates with a uniform angular velocity 20 rad s^–1 about its perpendicular bisector, in a uniform magnetic field 0.5 T. The direction of magnetic field is parallel to the axis of rotation. The potential difference between the two ends of the rod is ____V.

The correct answer is 0.

To understand why the potential difference between the two ends of the rod is zero, we can analyze the induced electromotive force (emf) developed in each half of the rotating rod.

Let the rod be denoted by AB, with its midpoint (perpendicular bisector) at O. The total length of the rod is:

$L=60 \mathrm{cm}=0.6 m$

The rod rotates about O, which means the length of each half, OA and OB, is:

$l=\frac{L}{2}=30 \mathrm{cm}=0.3 m$

When a rod of length $l$ rotates with a uniform angular velocity $\omega$ in a uniform magnetic field $B$ perpendicular to its plane of rotation (since the magnetic field is parallel to the axis of rotation, it is perpendicular to the length of the rod and its direction of motion), the induced emf between its axis of rotation (point O) and one of its outer ends (say point A) is given by:

$V_O-V_A=\frac{1}{2}B\omega l^2$

By symmetry, the induced emf between the center O and the other end B is exactly the same because both halves rotate in the same magnetic field with the same angular velocity:

$V_O-V_B=\frac{1}{2}B\omega l^2$

Subtracting these two equations to find the potential difference between the two ends A and B of the rod:

$(V_O-V_B)-(V_O-V_A)=\frac{1}{2}B\omega l^2-\frac{1}{2}B\omega l^2$

This simplifies to:

$V_A-V_B=0$

Thus, both ends of the rod are at the same electric potential, making the potential difference between them equal to 0 V.