Electricity and Magnetism
# Charges and Their Interactions

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.

$\displaystyle Q$ 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.

Let a chargeFind the **minimum** distance (in **m**) between the two charges.

**Details and Assumptions**

$\bullet$ $\displaystyle Q = 5\ \mu C$

$\bullet$ $\displaystyle q = 1\ \mu C$

$\bullet$ $\displaystyle d = 3\text{ cm}$

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

$\bullet$ $\displaystyle m = 1\text{ g}$

$\lambda$ 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.

The figure shows a non-conducting, non-uniformly charged ring with linear charge densitiesIf 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.
- $\lambda = \dfrac{\pi}{100} Cm^{-1}$
- $m = 1.5 kg$
- $R = 0.15 m$
- $|\vec{E}| = 1000 NC^{-1}$

$\displaystyle R$ 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**

$\bullet$ $\displaystyle q = 2\ \mu\text{C}$

$\bullet$ $\displaystyle R = 10\text{ cm}$

$\bullet$ $\displaystyle h = 15\text{ cm}$