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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.

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.

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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**

\(\bullet\) \(\displaystyle Q = 5\ \mu C\)

\(\bullet\) \(\displaystyle q = 1\ \mu C\)

\(\bullet\) \(\displaystyle d = 3\text{ cm}\)

\(\bullet\) \(\displaystyle v = 2 \text{ m/sec}\)

\(\bullet\) \(\displaystyle m = 1\text{ g}\)

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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**

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

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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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