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Electric fields describe the interaction of stationary charged matter. They underlie the working of diverse technology from atom smashers to the poor cell reception you're getting right now.

Find the **electrostatic force of interaction** between two parallel non conducting wires each of length \(l\) , charge density per unit length \(\lambda\), and separated by a distance \(d\).

Give your answer as the value of this force in \(\text{ Newtons }\) using \( \lambda = 10 \mu \text{C/ m}\), \( l = 1 \text{m} \), \(\text{ d = 1m }, \frac{1}{4 \pi \epsilon_{0} } = 9 \times 10^9\text{ N } m^2/ C^2 \)

**Note**:

The wires completely face each other (like if their equations were \(0 \leq x \leq L, y,z= 0\) and \(0 \leq x \leq L, y= d, z=0\))

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Consider a dielectric thin rod with total charge \(q=1~\mu\text{C}\) uniformly distributed over its length. The rod touches a grounded conducting sphere of radius \(R=50~\text{cm}\) as shown in the figure below. Find the charged induced in the sphere **in microcoulombs** if the length of the rod is \(L=1~\text{m}\). The following integral may be useful:
\[\int \frac{1}{\sqrt{a^{2}+x^{2}}}dx=\ln(x+\sqrt{x^2+a^{2}})+C.\]

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**in meters per second** of the charge located at C.

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An earthed metallic sphere is kept in a uniform electric field \( E_0 \hat{k} \). If \(V(r, \theta ,\phi ) \) denotes the potential function for the region outside the sphere then the value of \(\displaystyle \frac{V \left( 2R, \frac{\pi }{3} , \frac{ \pi }{2} \right) }{ V \left( 3R, \frac{-\pi }{3} , \frac{ 2\pi }{3} \right) } \) can be expressed as \( \dfrac{a}{b} \) for some positive coprime integers \(a,b\).
###### Image credit: Wikipedia Bob lonescu

Evaluate \(a+b\).

**Details and Assumptions:**

\(\hat{k} \) is the unit vector along the direction of \(z\) axis in a normal right handed Cartesian coordinate system.

Reference potential i.e. \( (V=0) \) is obviously the sphere since it is grounded.

\(R\) is the radius of the sphere.

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A particle \(P\) having a charge of \(-1\ \mu\text{C}\) and mass 2 g is held at a distance of \(x=1\text{ m}\) from the center of a disc, as shown in the figure above. The disc has a radius of 2 m and surface charge density \(6.84\times 10^{-4} \text{ C/m}^2\).

The particle is released, and flies toward the disc. Find the velocity (in **m/s**) of the particle when it has covered half the distance.

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