Wheatstone bridge principle is used to measure the specific resistance \( (S_1) \) of a given wire, having length \( L \), radius \( r \). If \( X \) is the resistance of the wire, then specific resistance is: $$ S_1 = X\left( \frac{\pi r^2}{L} \right). $$ If the length of the wire gets doubled, then the value of specific resistance will be:
- \( \frac{S_1}{4} \)
- \( 2S_1 \)
- \( \frac{S_1}{2} \)
- \( S_1 \)
The Correct Option is D
Approach Solution - 1
To solve this problem, we will begin by understanding the formula for specific resistance \( S_1 \) in terms of the physical dimensions of the wire and demonstrate how the length of the wire affects specific resistance.
The specific resistance \( S_1 \) of a wire is given by:
\(S_1 = X\left(\frac{\pi r^2}{L}\right)\)
Here:
- \( X \) is the resistance of the wire.
- \( r \) is the radius of the wire.
- \( L \) is the length of the wire.
The formula originates from the basic relation of resistance in terms of resistivity, length, and area of cross-section:
\(R = \rho \left(\frac{L}{A}\right)\)
where:
- \( \rho \) is the resistivity (specific resistance).
- \( A \) is the area of cross-section, given by \(\pi r^2\).
Now, let's analyze what happens if the length of the wire is doubled. The new length, \( L' \), is:
\(L' = 2L\)
Substituting for \( L' \) in the expression for specific resistance, we get a new specific resistance \( S_1' \):
\(S_1' = X\left(\frac{\pi r^2}{L'}\right)\)
Rewriting this with \( L' = 2L \):
\(S_1' = X\left(\frac{\pi r^2}{2L}\right)\)
However, we need to reconsider as specific resistance \( S_1 \) should be independent of length since \( S_1 \) strictly corresponds to the resistivity of the material, not dependent on geometrical changes once established initially.
The specific resistance, or resistivity, essentially remains the same regardless of doubling the length because it is a material property. Thus, \( S_1' = S_1 \).
Therefore, the correct answer is that the value of the specific resistance remains \(S_1\), which matches the given correct option:
\(S_1\)
Approach Solution -2
The specific resistance (or resistivity) of a material is defined as: \(Sl = X (\frac{πr^2}{L})\) ,
where X is the resistance, r is the radius, and L is the length of the wire. Specific resistance is a material property and does not change with changes in dimensions such as length or radius. Doubling the length of the wire affects the resistance X, but the specific resistance Sl remains unchanged.
The Correct answer is: \( S_1 \)
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