A metallic rod of length 1.0 ~m is pivoted at one end and rotated at a constant angular velocity of 20 ~rad/s in a uniform magnetic field of 0.5 ~T directed perpendicular to the plane of rotation. The free end of the rod slides on a circular conducting ring. If a resistor of 5 is connected between the pivot and the ring, what is the electrical power dissipated in the resistor?

Step-by-step solution

First, calculate the induced electromotive force (emf) across the ends of the rotating rod. The formula for the induced emf is: = 1 2 B L^2 Given values: B = 0.5 ~T = 20 ~rad/s L = 1.0 ~m Substituting the values: = 1 2 0.5 20 (1.0)^2 = 0.5 10 1 = 5 ~V Next, calculate the electrical power P dissipated in the resistor using the formula: P = ^2 R Given R = 5 : P = (5)^2 5 = 25 5 = 5 ~W If the 1 2 factor in the emf formula is omitted, the emf would be calculated as 10 ~V , leading to an incorrect power of 20 ~W . If th