Question

A wire of resistance X is cut into four equal pieces. These four pieces are connected in parallel. The resultant resistance of this combination is Y. The ratio X : Y is

• A. 20: 1
• B. 16: 1
• C. 1:4
• D. 4: 1

Answer 1

The correct option is B 16: 1

Answer is B.
The electrical resistance of a wire would be expected to be greater for a longer wire, less for a wire of larger cross sectional area, and would be expected to depend upon the material out of which the wire is made.The resistance of a wire can be expressed as $R=\rho \frac{L}{A}$,

where,
$\rho$ - Resistivity - the factor in the resistance which takes into account the nature of the material is the resistivity
L - Length of the conductor
A - Area of cross section of the conductor.

From this relation, we observe that the length is directly proportional to the resistance and the area of cross section is inversely proportional to the resistance.

In this case, the wire initially has the resistance $X=\rho \frac{L}{A}$.

Now, the length of the wire is cut into 4 equal pieces. So the length becomes L/4.

Therefore, the resistance Y of the cut wire is $Y=\rho \frac{\frac{L}{4}}{A}=1/4\left(\rho \frac{L}{A}\right),thatis,Y=\frac{X}{4}$.

The equivalent resistance when 4 such wires are connected in parallel is given as

$\frac{1}{Y}=\frac{1}{\frac{X}{4}}+\frac{1}{\frac{X}{4}}+\frac{1}{\frac{X}{4}}+\frac{1}{\frac{X}{4}}=\frac{4}{X}+\frac{4}{X}+\frac{4}{X}+\frac{4}{X}=4\left(\frac{4}{X}\right)=\frac{16}{X}$.

Hence, The ratio X: Y is 16:1.