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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.

Problem Solving

         

A homogeneous ball with mass \(m=1 ~\mbox{g}\) and total charge \(q=10^{-3}~\mbox{C}\) (uniformly distributed) is placed on a horizontal \(xy\)-plane. The ball starts rolling without slipping under the influence of a uniform electrostatic field \(\vec{E}=(E,0,0)\) with \(E=1~ \mbox{V/m}.\) Find the acceleration in meters per second squared of the center of mass of the ball.

If a point charge \( Q = -3 \text{ C } \) with mass \( m = 1 \text{ kg}\) is put at point \(P\) which is at a distance of \( z = 9 \text{ m}\) above the midpoint between two equal charges \( q = 25 \times 10^{-12} \text{ C }, \) at a distance of \( 2d =24 \text{ m}\) apart, the point charge will be move to point \(O.\) If the speed of the point charge at point \(O\) can be expressed as \( v = \sqrt{\frac{a}{b}} \text{ m/s}, \) where \(a\) and \(b\) are coprime positive integers, what is the value of \(a+b?\)

Let \( \pi \approx 3, \) and the permitivity of free space \( \epsilon_0 \approx 9\times 10^{-12} \frac{\text{C}^2}{\text{N*m}^2}. \)

There is a point charge with mass \( m = 1 \text{ kg}\) and charge \( q = -9\times 10^{-6} \text{ C } \) on a uniformly charged sphere with charge \( Q = 9\times 10^{-6} \text{ C } \) and radius \( R = 1 \text{ m}\). As shown in the above diagram, there is a hole on the sphere, so the point charge can move into the hole. If the period of the point charge can be expressed as \( T = 2\pi \sqrt{\frac{a}{b}} \text{ s}, \) where \(a\) and \(b\) are coprime positive integers, what is the value of \(a+b?\)

Let \( \pi \approx 3, \) and the permitivity of free space \( \epsilon_0 \approx 9\times 10^{-12} \frac{\text{C}^2}{\text{Nm}^2}. \)

A point particle with mass \( m = 1 \text{ kg}\) and charge \( q = -7\times 10^{-6} \text{ C } \) is rotating around another point particle with charge \( Q = 7\times 10^{-6} \text{ C } \),which is fixed at the center, with radius \( r = 2 \text{ m}\). If the speed of the circular motion can be expressed as \( v = \sqrt{\frac{a}{b}} \text{ m/s}, \) where \(a\) and \(b\) are coprime positive integers, what is the value of \(a+b?\)

Let \( \pi \approx 3, \) and the permitivity of free space \( \epsilon_0 \approx 9\times 10^{-12} \frac{\text{C}^2}{\text{N*m}^2}. \)

The potential at point \(P\) which is at a distance of \( z = 6 \text{ m}\) above the midpoint between two equal charges \( q = 500 \times 10^{-12} \text{ C } \), at a distance of \( 2d =16 \text{ m}\) apart, can be expressed as \( U = \frac{a}{b} \text{ J/C}, \) where \(a\) and \(b\) are coprime positive integers. What is the value of \(a+b?\)

Let \( \pi \approx 3, \) and the permitivity of free space \( \epsilon_0 \approx 9\times 10^{-12} \frac{\text{C}^2}{\text{N*m}^2}. \)

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