Question

A thin plastic rod of length 2.6 m is rubbed all over with wool, and acquires a charge of 98 nC, distributed uniformly over its surface. Calculate the magnitude of the electric field due to the rod at a location 13 cm from the midpoint of the rod. Do the calculation two ways, first using the exact formula for a rod of any length, and second using the approximate formula for a long rod.

Answers

  1. Answer:

    By exact formula

    5076.59N/C

    And by approximation formula

    5218.93N/C

    Explanation:

    We are given that

    Length of rod,L=2.6 m

    Charge,q=98nC=98\times 10^{-9} C

    1nC=10^{-9} C

    a=13 cm=0.13 m

    1 m=100 cm

    By exact formula

    The magnitude of  the electric field due to the rod at a location 13 cm from the midpoint of the rod=\frac{kq}{a}\times \frac{1}{\sqrt{a^2+\frac{L^2}{4}}}

    Where k=9\times 10^9

    Using the formula

    The magnitude of  the electric field due to the rod at a location 13 cm from the midpoint of the rod=\frac{9\times 10^9\times 98\times 10^{-9}}{0.13}\times \frac{1}{\sqrt{(0.13)^2+\frac{(2.6)^2}{4}}}=5076.59N/C

    In approximation formula

    a<<L

    a^2+(\frac{L}{2})^2=\frac{L^2}{4}

    Therefore,the magnitude of  the electric field due to the rod at a location 13 cm from the midpoint of the rod=\frac{kq}{a}\times \frac{1}{\sqrt{\frac{L^2}{4}}}

    The magnitude of  the electric field due to the rod at a location 13 cm from the midpoint of the rod=\frac{9\times 10^9\times 98\times 10^{-9}}{0.13}\times \frac{1}{\sqrt{\frac{(2.6)^2}{4}}}=5218.93N/C

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