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Electric Fields: Level 4-5 Challenges


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 \( \lambda = 10 \mu \text{C/ m}\), \( l = 1 \text{m} \), \(\text{ d = 1m }, \frac{1}{4 \pi \epsilon_{0} } = 9 \times 10^9\text{ N } m^2/ C^2 \)


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

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: \[\int \frac{1}{\sqrt{a^{2}+x^{2}}}dx=\ln(x+\sqrt{x^2+a^{2}})+C.\]

Three identical point charges, each with mass \(m=1~\text{g}\), are connected by strings of the same length \(l\) so that they form an equilateral triangle ABC. Initially, the charges are at rest and the electrostatic potential energy of the system is \(U_{0}=1~\text{J}\). Suddenly, the string connecting the charges at A and B snaps. What will be the maximum speed in meters per second of the charge located at C.

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 \(a,b\).

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.

Image credit: Wikipedia Bob lonescu

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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