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Charges and Their Interactions

The motion of charged matter underlies many things we enjoy like phones and plasma globes. It also puts a permanent end to the enjoyment of some 6,000 people per year, as fatal lightning strikes.

Level 3-4

         

A particle of mass \(m=1\text{ kg}\) and charge \(q=3\text{ C}\) is hung from a light inextensible string of length \(l=2\text{ m}\). The entire system is placed in a uniform horizontal electric field of magnitude \(8\text{ V/m}\). What must be the minimum initial horizontal velocity with which the particle must be projected so that it completes a vertical circle?

Details and Assumptions

  • The gravitational acceleration is \(g=-10\text{ m/s}^2\).
  • The particle is initially at the lowest point.

Consider two parallel, grounded, conducting plates located at \(x=-L\) and \(x=L\). A charge of \(Q=-92 \space \mu\text{C}\) is placed at \((L/2,0,0)\). What is the total charge induced on the plate located at \(x=L\) in microcoulombs?

Let a charge \(\displaystyle Q\) be held fixed in space. Another charge \(\displaystyle q\) of mass \(m\) is thrown with a velocity of \(\displaystyle v\) from an infinitely far place, as shown in the figure, such that it is at a separation of \(\displaystyle d\) initially.

Find the minimum distance (in m) between the two charges.

Details and Assumptions

\(\bullet\) \(\displaystyle Q = 5\ \mu C\)
\(\bullet\) \(\displaystyle q = 1\ \mu C\)
\(\bullet\) \(\displaystyle d = 3\text{ cm}\)
\(\bullet\) \(\displaystyle v = 2 \text{ m/sec}\)
\(\bullet\) \(\displaystyle m = 1\text{ g}\)

The figure shows a non-conducting, non-uniformly charged ring with linear charge densities \(\lambda\) and \(- \lambda\), mass \(m\) and radius \(R\). This ring is placed on a rough horizontal surface and a horizontal electric field \(\vec{E}\) is switched on.

If at some instant, the ring is in the position shown above and is rolling without slipping, find the minimum coefficient of friction \(\mu_{min}\) required.

Details & Assumptions

  1. A uniform gravitational field \(|\vec{g}| = 10 ms^{-2}\) acts downward.
  2. \(\lambda = \dfrac{\pi}{100} Cm^{-1}\)
  3. \(m = 1.5 kg\)
  4. \(R = 0.15 m\)
  5. \(|\vec{E}| = 1000 NC^{-1}\)

A thin ring of radius \(\displaystyle R\) is charged with a charge \(\displaystyle q\). It is then placed over an infinitely large grounded conducting plane at a height \(\displaystyle h\), as shown in the figure above.

Find the magnitude of the surface charge density \(\displaystyle \sigma\) (in C/m\(^2\)) of the induced charge at the point on the plane which is exactly below the ring's center.

Details and Assumptions

\(\bullet\) \(\displaystyle q = 2\ \mu\text{C}\)
\(\bullet\) \(\displaystyle R = 10\text{ cm}\)
\(\bullet\) \(\displaystyle h = 15\text{ cm}\)

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