Question

    A metallic conductor of circular cross-section has

    radius R. If current density varies as J = Jₒ(1−r/R), the total current through the conductor is:
    A (π RJₒ) / 6 Correct Answer Incorrect Answer
    B (π RJₒ) / 4 Correct Answer Incorrect Answer
    C (π R²Jₒ) / 3 Correct Answer Incorrect Answer
    D (π R²Jₒ) / 2 Correct Answer Incorrect Answer

    Solution

    The current density J is given as a function of the radial distance r from the center of the circular cross-section: J = Jₒ(1−r/R) where Jₒ is a constant and R is the radius of the conductor.   To find the total current I through the conductor, we need to integrate the current density over the cross-sectional area of the conductor. Consider a small circular area element dA at a radial distance r from the center. The area of this element is dA = 2πrdr. The current dI through this area element is given by dI = JdA. dI = Jₒ(1−r/R) (2πrdr) To find the total current I, we integrate dI over the entire cross-section of the conductor, from r = 0 to r = R:

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