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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.

An electron floats at rest above a sheet of charge with uniform charge density \(\sigma\). What is \(\sigma\) in **\(\mbox{C/m}^2\)**?

**Details and assumptions**

- The mass of the electron is \(9.1 \times 10^{-31}~\mbox{kg}\).
- The electric charge on the electron is \(-1.6 \times 10^{-19}~\mbox{C}\).
- The acceleration due to gravity is \(-9.8~\mbox{m/s}^2\).
- The vacuum permittivity is \(\epsilon_0=8.85 \times 10^{-12}~\mbox{F/m}\).

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A point particle with mass \( m = 2 \text{ kg}\) and charge \( Q = 33 \text{ C } \) is shot with speed \( v = 8 \text{ m/s}\) at an angle of \(\theta = 30^o \) with the ground. There is an electric field \( \vec{E} = -22 \hat{y}\text{ N/C}\) heading perpendicularly to the ground. When the particle falls to the ground, the distance from the original point can be expressed as \( R = \frac{a}{b} \text{ m}, \) where \(a\) and \(b\) are coprime positive integers, what is the value of \(a+b?\)

Assume that \(\sqrt{3} = 1.5 \)

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A point particle with mass \( m = 1\text{ kg}\) and charge \( Q = 8 \text{ C } \) is shot with speed \( v = 3 \text{ m/s}\) at an angle of \(\theta = 30^\circ \) with the ground. There is an electric field \( \vec{E} = 3\hat{x} - 6\hat{y} \text{ N/C},\) as shown in the above diagram. When the particle falls to the ground, the distance from the original point can be expressed as \( R = \frac{a}{b} \text{ m}, \) where \(a\) and \(b\) are coprime positive integers. What is the value of \(a+b?\)

Assume that \(\sqrt{3} = 1.5 .\)

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A point particle with mass \( m = 6 \text{ kg}\) and charge \( Q = 6 \text{ C } \) is shot with initial speed \( v(0) = 6 \text{ m/s}\) at an angle of \(\theta = 60^\circ \) with the ground. There is an electric field \( \vec{E} = -6 \hat{y}\text{ N/C},\) as shown in the above diagram. When \(t = \frac{3}{4}\text{ s},\) what is the speed of the particle ?

Assume that \(\sqrt{3} = 1.5 \)

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A point particle with mass \( m = 1 \text{ kg}\) and charge \( Q = 9 \text{ C } \) is shot with speed \( v = 6 \text{ m/s}\) at an angle of \(\theta = 30^o \) with the ground. There is an electric field \( \vec{E} = -9 \hat{y}\text{ N/C},\) as shown in the above diagram. If the maximum height can be expressed as \( H = \frac{a}{b} \text{ m}, \) where \(a\) and \(b\) are coprime positive integers, what is the value of \(a+b?\)

There is no gravitational field present.

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