A uniform heating wire of resistance R produces thermal energy H in time t when connected directly across a constant voltage battery. The wire is now cut into three equal pieces, and these pieces are connected in parallel across the same battery. The thermal energy produced by this parallel combination in the same time t will be

Step-by-step solution

The initial thermal energy produced by the wire is given by H = V^2 R t . When the uniform wire is cut into three equal pieces, the resistance of each piece becomes R 3 because resistance is directly proportional to length. When these three pieces are connected in parallel, their equivalent resistance R_ eq is calculated as: 1 R_ eq = 1 R/3 + 1 R/3 + 1 R/3 = 3 R + 3 R + 3 R = 9 R Thus, R_ eq = R 9 . The new thermal energy produced in the same time t across the same battery is: H' = V^2 R_ eq t = V^2 R/9 t = 9 ( V^2