Question

FIGURE is a cross section of two infinite lines of charge that extend out of the page. The linear charge densities are $\pm \lambda$. Find an expression for the electric field strength $E$at height $y$above the midpoint between the lines.

Short answer

The expression for electrical field strength, $E=\frac{2\lambda d}{\pi {ϵ}_o\left(d^2+4y^2\right)}.$

Step-by-step solution

Step1:  Definition of Electrical charge:

Charges that are similar repel one other, while charges that are dissimilar attract one other. when matter is placed in an electromagnetic field, an electric charge, which causes it to experience a force. A positive or negative charge can exist (commonly carried by protons and electrons respectively).

Find the Electrical charge

Electrical field make to horizontal component by infinite lines of charge density $\lambda$

$E_x=2E_x\sin \theta$

$\left(\sin \theta =\frac{d}{2\sqrt{\frac{d^2}{4}+y^2}}\right)$

$E=\frac{\lambda }{2\pi {ϵ}_{o}l}$

$\left(l=\sqrt{\frac{d^2}{4}+y^2}\right)$

$E_x=2\times \frac{\lambda }{2\pi {ϵ}_{o}l}\times \frac{d}{2l}$

$E=\frac{\lambda d}{\pi {ϵ}_{o}2\times \frac{1}{4}\left(d^2+y^{2}4\right)}$

$E=\frac{2\lambda d}{\pi {ϵ}_o\left(d^2+4y^2\right)}$