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A charged wire of negligible thickness has length 2L units and has a linear charge density λ. Consider the electric field E-vector at the point P, a distance d above the midpoint of the wire. The field E-vector points along one of the primary axes, yWhat is the magnitude E of the electric field at point P? Throughout this part, express your answers in terms of the constant k, defined by k=1/(4πε)

Answer :

Answer:

[tex]E=2K\lambda d\dfrac{L }{d^2\sqrt{L^2+d^2}}[/tex]

Explanation:

Given that

Length= 2L

Linear charge density=λ

Distance= d

K=1/(4πε)

The electric field at point P

[tex]E=2K\int_{0}^{L}\dfrac{\lambda }{r^2}dx\ sin\theta[/tex]

[tex]sin\theta =\dfrac{d}{\sqrt{d^2+x^2}}[/tex]

[tex]r^2=d^2+x^2[/tex]

So

[tex]E=2K\lambda d\int_{0}^{L}\dfrac{dx }{(x^2+d^2)^{\frac{3}{2}}}[/tex]

Now by integrating above equation

[tex]E=2K\lambda d\dfrac{L }{d^2\sqrt{L^2+d^2}}[/tex]

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