Physics

Problem

A rectangular circuit board contains two parallel branches that rejoin at a common node, forming a single closed rectangular loop ABCDA (in that order). Starting at node A and moving clockwise, the elements encountered along the loop are: 1. Branch AB (upper branch): an ideal battery of labeled emf

12.0\,\text{V}

in series with a carbon resistor

R_1

. 2. Branch BC (right side): a segment of printed copper conductor of negligible resistance. 3. Branch CD (lower branch): a wire-wound resistor

R_2

whose resistance is stable under load. 4. Branch DA (left side): a copper conductor of negligible resistance. With the circuit operating in steady state, a technician measures potential differences along the loop in the clockwise direction, obtaining the data shown in the table. Each measurement carries an uncertainty of

\pm0.2\,\text{V}

Segment	Measured $\Delta V$ (V)
Across battery	$+12.1$
Across $R_1$	$-4.6$
Across BC trace	$-0.3$
Across $R_2$	$-7.7$
Across DA trace	$+0.2$

(a) Compute the algebraic sum of the measured potential differences for one complete traversal of the loop (A→B→C→D→A). Include the appropriate combined uncertainty for the sum, assuming independent random errors. (b) Determine whether the measurements satisfy energy conservation at the 95 percent confidence level. State your criterion clearly and justify your conclusion quantitatively. (c) If the loop fails the test in part (b), identify which single measurement, if adjusted, would most effectively restore agreement. Calculate the adjusted value required, and explain your choice.

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