Question:medium

Potential difference at the ends of a conductor of length \(l\) is \(V\). Diameter of the conductor is \(d\). What will be the effect on the electric field inside the conductor and its resistance, if (i) \(V\) is halved? (ii) \(l\) is doubled? (iii) \(d\) is doubled?

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Use E = V/l (independent of diameter) and R = 4 rho l/(pi d^2) (independent of V). Then scale each quantity for the change asked.
Updated On: Jul 10, 2026
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Solution and Explanation

Step 1: Set up proportionalities.
Inside the wire the field is the potential gradient, \(E = V/l\), so \(E \propto V\) and \(E \propto 1/l\), while \(E\) is independent of the diameter. The resistance is \(R = \rho l / A\) with \(A = \pi d^2/4\), giving \(R = 4\rho l/(\pi d^2)\), so \(R \propto l\), \(R \propto 1/d^2\), and \(R\) is independent of \(V\). I will track only the ratios.

Step 2: Halving V.
Since \(E \propto V\), halving \(V\) halves the field: \(E \to E/2\). Since \(R\) does not contain \(V\) at all, the resistance stays the same. So field halves, resistance constant.

Step 3: Doubling l.
Since \(E \propto 1/l\), doubling \(l\) makes \(E \to E/2\). Since \(R \propto l\), doubling \(l\) makes \(R \to 2R\). So field halves and resistance doubles.

Step 4: Doubling d.
\(E\) has no \(d\) dependence, so the field is unaffected. Since \(R \propto 1/d^2\), replacing \(d\) by \(2d\) multiplies \(R\) by \(1/(2)^2 = 1/4\), so \(R \to R/4\). Field unchanged, resistance quartered.

Step 5: Collected results.
\[\boxed{(i)\ E \to \tfrac{E}{2},\ R\ \text{same};\quad (ii)\ E \to \tfrac{E}{2},\ R \to 2R;\quad (iii)\ E\ \text{same},\ R \to \tfrac{R}{4}}\]
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