What is the ratio of the magnitude of the gravitational force compared to the electric force from one proton on the other? Is this a big difference? Why is the electric force more viable as a means of holding an atom together compared to the gravitational force?
Suppose a positive and negative charge are separated by a distance \(d \) and lie on the \(x\)-axis equidistant from the origin with the positive charge at \(x=-d/2\) and the negative charge at \(x=d/2\text{.}\) A positive point charge \(+Q \) is placed a distance \(r \) from the center of the dipole on the positive \(x\)-axis.
Overlaid on the appropriate coordinate system, sketch a diagram of the physical situation and label all quantities of interest.
Construct an expression for the net force exerted on the dipole by charge \(Q \text{.}\) Be sure to use unit vector notation when constructing your equation.
Explain: Do you expect the net force be toward \(Q \) or away from \(Q \text{?}\) Does your expression from part B confirm this? Use sentences, symbols and diagrams to back up your answer.
Suppose that the lengths \(r \gg d \) as is usually the case for atomic dipoles. Use the binomial expansion as an approximation tool to approximate the force to first order. Using the binomial approximation, simplify your expression for the net force exerted on the dipole from charge \(Q \text{.}\)
Explain: How can this force have an inverse-cube dependence on \(r\) ? Doesn’t Coulomb’s law assert that the electric force depends on the inverse square of distance? Explain why this is possible in this situation.
Sensemake: What happens to the value of the force as \(d \) gets very small? What happens to the force as \(d \rightarrow 0\text{?}\) What does \(d \rightarrow 0\) mean physically for a dipole?
A piece of pollen typically has both a small mass (about \(5 \mathrm{~ng}\)) and a small amount of charge (about \(+1 \mathrm{~fC}\)). Estimate the electric field that a honeybee needs to create to carry a typical piece of pollen.
In a simple model of the hydrogen atom, the electron moves in a circular orbit of a radius \(0.053 \mathrm{~nm}\) around a stationary proton. How many revolutions per second does the electron make?
Two \(5.0 \mathrm{~g}\) point charges on \(1.0 \mathrm{~m}\) long threads are charged to \(+100.0 \mathrm{~nC}\text{.}\) Determine the unknown angle \(\theta\text{.}\) You can assume that \(\theta\) is a small angle.
An ideal spring has spring constant \(k_s\) (to distinguish it from the electrostatic constant \(k\)) and equilibrium length \(l\text{.}\) Then, you glue two identical negative point charges to the ends of the spring and observe that the equilibrium length doubles. Determine the amount of charge on each end of the spring.
A system of protons and electrons has a net charge of \(-1.12 \times 10^{-18} \mathrm{~C}\text{.}\) If the system contains 15 protons, how many electrons are in the system?
Consider a cotton sock inside a dryer machine whose drum (inner container) is made of steel. After tumbling around in the dryer’s drum for a while, the cotton sock has a net charge on it now. Which of the following charge transfers most likely happened in this sock-drum system?
Consider a positive charge \(q_1 = +4 .00 \mathrm{~nC}\) that exists in a space far away from other charges. What is the electric field at a point P in space that is \(15.0 \mathrm{~m}\) to the left of this point charge?
An electron and a proton are \(3.00 \mathrm{~\mu m}\) apart from each other. What is the magnitude of the electric force by the proton on the electron?
Consider two small, charged spheres, \(q_1\) and \(q_2\text{,}\) a distance \(\Delta x\) apart.
If \(\Delta x\) increases by a factor of \(16/9\text{,}\) and \(q_1\) and \(q_2\) remain the same, by what factor does the force between the two change?
If \(\Delta x\) increases by a factor of \(4/\sqrt{2}\text{,}\)\(q_1\) increases by a factor of \(2\text{,}\) and \(q_2\) increases by a factor of \(\sqrt{2}\text{,}\) by what factor does the force between the two change?
A \(-3.00 \mathrm{~C}\) charge is placed in an external electric field. It is observed that the electric force on this charge is \(\vec{F} = 9 \hat{x} + 3 \hat{y} -6 \hat{z} \mathrm{~N}\text{.}\) What is the external electric field?
An electron is traveling in the negative \(x\)-direction when an external electric field pointing in the positive \(x\)-direction is turned on in that region. Which of the following statements are true about the electron right after the field is turned on?
An electron is traveling with a velocity that has equal positive \(x\) and \(y\) components. Suddenly an electric field is turned on in the positive \(y\)-direction in this region. Which of the following statements are true about the electron right after the field is turned on?
A 4.00 kg object with a net charge of \(-2.00 \mathrm{~C}\) is placed at rest above the surface of the Earth. There also exists a uniform external electric field of magnitude \(3.00 \mathrm{~N/C}\) pointing directly downward. When released, what is the magnitude of the acceleration of this object?
Two small balls, each of mass \(5.0 \mathrm{~g}\text{,}\) are attached to silk threads \(50 \mathrm{~cm}\) long, which are in turn tied to the same point on the ceiling, as shown above. When the balls are given the same charge \(Q\text{,}\) the threads hang at \(5.0^\circ\) to the vertical. What is the magnitude of \(Q\text{?}\) What are the signs of the two charges?
