Answer to Question #203060 in Electric Circuits for Nabeel khan

Question #203060

Two charges 2C and -2C are at 0.01m. Calculate the electric field intensity 0.3m in equatorial plane.


1
Expert's answer
2021-06-07T08:30:35-0400


Let us consider two equal and opposite charges qq and q-q at points A and B separated by a distance 2d2d as shown in the figure. We have to find the electric field at an equatorial point PP distant xx from the center OO .

From ΔAOP\Delta AOP and ΔBOP\Delta BOP ,

AP=BP=x2+d2AP=BP=\sqrt{x^2+d^2}

The electric field at P due to qq is

Ea=14πϵ0qx2+d2\vec{E}_a=\frac{1}{4\pi \epsilon_0}\frac{q}{x^2+d^2} directed along AP.

The electric field at P due to q-q is

Eb=14πϵ0qx2+d2\vec{E}_b=\frac{1}{4\pi \epsilon_0}\frac{q}{x^2+d^2} directed along PB.

Note that the electric fields Ea\vec{E}_a and Eb\vec{E}_b have equal magnitude i.e,

Ea=Eb=E(say)|\vec{E}_a|=|\vec{E}_b|=E(say)

To find the resultant electric field at PP we will resolve Ea\vec{E}_a and Eb\vec{E}_b in horizontal and vertical components as shown below.



The vertical components EasinθE_a\sin\theta and EbsinθE_b\sin\theta will cancel each other (since Ea=Eb|\vec{E}_a|=|\vec{E}_b| ).

The horizontal components EacosθE_a\cos\theta and EbcosθE_b\cos\theta will add up to give the resultant electric field.

\therefore Resultant electric field at PP is

Er=Eacosθ+EbcosθEr=2EcosθEr=214πϵ0qx2+d2.dx2+d2Er=14πϵ0q.2d(x2+d2)3/2E_r=E_a\cos\theta+E_b\cos\theta\\ \Rightarrow E_r=2E\cos\theta\\ \Rightarrow E_r=2\frac{1}{4\pi\epsilon_0}\frac{q}{x^2+d^2}.\frac{d}{\sqrt{x^2+d^2}}\\ \Rightarrow \boxed{ E_r=\frac{1}{4\pi\epsilon_0}\frac{q.2d}{(x^2+d^2)^{3/2}}} directed along AB.

Given, q=2C,2d=0.01 m,x=0.3 mq=2C, 2d=0.01\ m, x=0.3\ m

We know 14πϵ0=9×109N.m2.C2\frac{1}{4\pi\epsilon_0}=9\times 10^9 N.m^2.C^{-2}

Er=9×109×2×0.01(0.32+0.0052)3/2 N/CEr=6.7×109 N/C\therefore E_r=9\times 10^9\times \frac{2\times 0.01}{(0.3^2+0.005^2)^{3/2}}\ N/C\\ E_r=6.7\times 10^9\ N/C


Answer: The electric filed intensity at 0.3m in equatorial plane is 6.7×109 N/C6.7\times 10^9 \ N/C



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