The Physics
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Opus in profectus

# Electric Potential

## Problems

### practice

1. A charge of -1.0 μC is located on the y-axis 1.0 m from the origin at the coordinates (0,1) while a second charge of +1.0 μC is located on the x-axis 1.0 m from the origin at the coordinates (1,0). Determine the value of the following quantities at the origin…
1. the magnitude of the electric field,
2. the direction of the electric field,
3. the electric potential (assuming the potential is zero at infinite distance), and
4. the energy needed to bring a +1.0 μC charge to this position from infinitely far away.
2. A proton (mass m, charge +e) and an alpha particle (mass 4m, charge +2e) approach one another with the same initial speed v from an initially large distance. How close will these two particles get to one another before turning around?
3. sketch-v.pdf
The diagram below shows the location and charge of four identical small spheres. Find the electric potential at the five points indicated with open circles. Use these results and symmetry to find the potential at as many points as possible without additional calculation. Write your results on or near the points. Sketch at least 4 equipotential lines. Pick round values seperated by a uniform interval. At least one of the lines should be disconnected.

4. Fission is the splitting of a heavy atomic nucleus into two roughly equal halves accompanied by the release of a large amount of energy. An atomic nucleus can be modeled as a sphere whose charge is distributed uniformly across its entire volume. Determine the energy released when a heavy nucleus undergoes nuclear fission using electrostatic principles.
1. Derive an equation for the electrostatic energy needed to assemble a charged sphere from an infinite swarm of infinitesimal charges located infinitely far away. (In other words, use calculus.) Let R be the sphere's radius, Q be its total charge, V be its volume, and ρ be its charge density.
2. Express the total energy of two half-sized spheres in terms of the energy of one whole sphere. Half-sized spheres have half the volume and half the charge of a whole sphere (because charge density is assumed to be constant).
3. Calculate the energy released when a nucleus of uranium 235 (the isotope responsible for powering some nuclear reactors and nuclear weapons) splits into two identical daughter nuclei. Give your final answer in the preferred unit for nuclear reactions, the megaelectronvolt. (A nucleus of 23592U has a radius of 5.8337 fm.)

### conceptual

1. In a region where the electric field is constant, as it is between two oppositely charged parallel plates, is the voltage also constant? Explain your answer.
2. Two related questions.
1. What do the electric fields lines look like in a region where the magnitude of the electric field is uniform?
2. What do they look like in a region where the electric potential is uniform?

### numerical

1. The inside of a human nerve cell is more negative than the outside by about −70 mV. When a nerve impulse propagates down an axon, the polarity reverses and the inside is more positive than the outside by +40 mV. This action potential lasts 2 ms and then the original resting potential is restored. All of this takes place in the space of about 6 nm, the thickness of the cell membrane.
1. What are the magnitude and direction of the electric field (in V/m) across the membrane of a neuron during…
1. the resting phase
2. the action phase
2. How much work is done moving a single sodium ion (Na+) across the cell membrane of a neuron? State your answer in…
1. joules
2. electronvolts
3. What is the power of this microscopic event?

### statistical

1. millikan.txt
The data in this file show the charge on an oil drop determined by Millikan during one particular run of his famous experiment (1911). What value for the elementary charge, e, can be deduced from this data? Show the work used to arrive at your answer. Source: Halliday, David & Robert Resnick. Physics: Parts 1&2. 3rd Edition. New York: Wiley, 1978: 600.
2. oil-drop.txt
The data in the accompanying text file were adapted from a paper written by Robert Millikan in 1911. The first column gives the trial number, the second column gives the battery voltage, and the last two columns give the time for the oil drop to fall down and then rise up between the cross hairs on the observing window. The table below provides the additional data needed to complete this assignment.
Constants in Millikan's experiment
quantity value
distance between charged plates 1.600 cm
distance between cross hairs 1.010 cm
viscosity of air at 25.2 °C 18.36 μPa s
density of air at 300 K 3.556 kg/m3
density of oil at 25 °C 896.0 kg/m3
acceleration due to gravity 9.81 m/s2
Using a spreadsheet program or similar data analysis software, determine the values of the following quantities and add them to the table.
1. the speed of the drop on the way down
2. the speed of the drop on the way up
3. the radius of the drop
4. the mass of the drop
5. the strength of the electric field
6. the charge on the drop
7. the number of elementary charges
8. the magnitude of the elementary charge