Question

The electric resistance of a certain wire of iron is R. If its length and radius are both doubled then

• A. The resistance will be halved and the specific resistance will remain unchanged
• B. The resistance will be halved and the specific resistance will be doubled
• C. The resistance and the specific resistance, will both remain unchanged
• D. The resistance will be doubled and the specific resistance will be halved

Answer 1

The Correct Option is A

To solve the problem, we need to understand how the resistance of a wire changes with its physical dimensions, namely its length and radius.

The resistance R of a wire is given by the formula:

R = \rho \frac{L}{A}

where:

• \rho is the resistivity (also called specific resistance),
• L is the length of the wire,
• A is the cross-sectional area of the wire.

The cross-sectional area A of a wire with radius r is given by A = \pi r^2.

Let's analyze how the resistance changes when both the length and radius of the wire are doubled:

1. If the length is doubled, L\_2 = 2L.
2. If the radius is doubled, the new radius r\_2 = 2r, and the new cross-sectional area becomes: A\_2 = \pi (2r)^2 = 4\pi r^2 = 4A.
3. Substituting the new length and area, the new resistance R\_2 will be: \[ R\_2 = \rho \frac{L\_2}{A\_2} = \rho \frac{2L}{4A} = \frac{1}{2}\rho \frac{L}{A} = \frac{R}{2} \] Hence, the resistance is halved.

Now, about the specific resistance, also known as resistivity (\rho):

• The specific resistance is a material property and remains unchanged under changes to the physical dimensions of the wire, such as length and radius.

Thus, the correct option is: The resistance will be halved and the specific resistance will remain unchanged.