Question

A constant voltage is applied between two ends of a uniform metallic wire. Some heat is developed in it. The heat developed is doubled, if

• A. both the length and radius of the wire are halved
• B. both the length and radius of the wire are doubled
• C. the radius of the wire is doubled
• D. the length of the wire is doubled

Hint:

Heat developed in a conductor is proportional to the square of the current and the resistance. To double the heat, both the length and the radius must be doubled to increase the resistance.

Answer 1

The Correct Option is B

The heat developed in a wire is given by Joule’s law: \[ H = I^2 R t \] where \( H \) is the heat developed, \( I \) is the current, \( R \) is the resistance, and \( t \) is the time. The resistance of a wire is given by: \[ R = \rho \frac{L}{A} \] where \( \rho \) is the resistivity, \( L \) is the length, and \( A \) is the cross-sectional area of the wire. For a wire with radius \( r \), the cross-sectional area is \( A = \pi r^2 \). To double the heat developed, we need to double the resistance. Since resistance is directly proportional to the length and inversely proportional to the cross-sectional area, doubling both the length and radius of the wire will double the resistance, thus doubling the heat developed. 
Hence, the correct answer is (b).