Question
A uniform wire is bent into a circle and a battery is
connected across a chord that subtends an angle of 60Ā° at the center. What is the ratio of current in the shorter arc to that in the longer arc?Solution
Let the total resistance of the circular wire be R . Since the wire is uniform, the resistance is proportional to the length. The total angle subtended at the center by the circle is 360 Ā° Ā or 2Ļ radians. The chord subtends an angle of 60 Ā° Ā at the center. The shorter arc subtends an angle of 60 Ā° Ā at the center. Its length is (60 Ā° /3 60 ) Ć ( circumference ) = (1/6) ā Ć 2Ļr . The resistance of the shorter arc Rā Ā = (60 Ā°/3 60 ) Ć R = R/6 The longer arc subtends an angle of 360 Ā° Ā - 60 Ā° = 300 Ā° Ā at the center. Its length is (300 Ā° /3 60 ) Ć R = 5R/6 The battery is connected across the chord, which means the shorter arc and the longer arc are in parallel across the battery. Let the potential difference across the battery (and hence across both arcs) be V. 
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