# Gradient of potential energy equals minus the force

1) Both the Coulomb law and gravitational forces lead to potential energies of the form [see the attached file for full equations and symbols] where [...] denotes [...] in the case of the Coulomb force and [...] for gravity, and r1 and r2 are the positions of the two particles. Show in detail that [...] is the force on particle 1 and [...] that on particle 2.

2) Write out the arguments of all the potential energies of the four-particle system in

[...]

For instance U = U (r1, r2, ... , r4), whereas U_34 = U_34 (r3 ??" r4). Show in detail that the net force on particle 3 (for instance) is given by [...].

Please see the attachment for the full questions.

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#### Solution Preview

You can do problem 1) in Cartesian coordinates or in spherical coordinates. Let me give an outline of how you can do it in Cartesian coordinates first.

The potential energy is:

U = gamma/|r1 - r2|

We will calculate minus the gradient and then compare that with the force law to see if minus the gradient is indeed the force. In Cartesian coordinates you can write:

|r1 - r2| = sqrt[(x1-x2)^2 + (y1 - y2)^2 + (z1 - z2)^2]

The components of the gradient are the derivatives with respect to the coordinates. You can take the gradient w.r.t. to r1 = (x1,y1,z1) or r2 = (x2,y2,z2). If we calculate the gradient w.r.t. r1 and take the x-component then we have to differentiate U w.r.t. x1. Let's first rewrite U as:

U = gamma [(x1-x2)^2 + (y1 - y2)^2 + (z1 - z2)^2]^(-1/2)

dU/dx1 = -1/2gamma[(x1-x2)^2 + (y1 - y2)^2 + (z1 - z2)^2]^(-3/2)*2(x1-x2) =

-gamma[(x1-x2)^2 + (y1 - y2)^2 + (z1 - z2)^2]^(-3/2)*(x1-x2).

The term in the square brackets is |r1-r2|^2, so yo can write this in a "mixed" form as:

dU/x1 = -gamma/|r1-r2|^3 (x1-x2) ...

Electrostatic Energy

1. A negative charge in an electric field moves from a point where the potential is zero to a point whre it is -100V. Discuss the energy change and the work done.

2. Imagine a hollow, spherical, positively charged conductor of radius R that is far away from any other bodies. Draw a graph of its potential V as a function of r out from its center. Discuss your results.

3. With question 2 in mind, suppose we place a neutral hollow conducting sphere in the vicinity of the charged sphere. Draw a rough graph of the potential along the center line of both spheres, and compare it to the previous potential without he neutral conductor. Discuss your results.

4. Describe the potential of a negative point-charge as a function of distance r. Explain your thinking.

5. A positive point-charge is located a short distance above a large conducting horizontal plane. Describe the field lines and equipotentials. Compare your results with that of an electric dipole. Explain your conclusions.

6. Imagine a square with point-charges at each corner. At the ends of one diagonal, the charges are +q; at the ends of the other they are -q. This arrangement is called and electric quadrupole. Make a rough sketch of the field lines and equipotentials and explain your reasoning.

7. It is sometimes desirable, especially in high-voltage applications, to replace a capacitor by an equivalent string of several capacitors in series. Discuss the possible reasons for this. How do the sized of the several series capacitors compare to the original single one?

8. Multiple choice question

In the case of a nonconductor: (a) its surface must be at a single potential (b) its entire volume, except for the surface, is at a constant potential, (c) different regions may be at different potentials or (d) the potential must be zero everywhere within it.

Explain your choice.

9. Electric field lines always point toward: (a) ground (b) a region of higher potential (c) a region of lower potential (d) positive charge (e) none of these. Explain your choice.

10. The potential as we get closer and closer to a point-charge: (a) approaches +-infiniity (b)zero (c) indeterminate (d) is exceedingly small, but not zero (e) none of these. Explain your choice

11. Given a group of nearby charges whose net value is nonzero, the equipptensial surface at a very great distance is: (a) nearly a plane (b) nearly a sphere (c) quite indeterminate (d) nearyl a cylinder (e) none

Explain your choice

12. Any closed equipotential surface that does not surround a net charge must: (a) be at zero-potential (b) be a sphere (c) enclose an equipotential volume (d) be infinately small (e) none. Explain your choice.

13. A volume of space is found to have a constant poteintial everywhere within it. It follows tht in the region: (a)the E-field is zero (b) the potential is zero (c) the E-field is finite and uniform (d) the potential gradient is a nonzero constant (e) none of these. Explain your choice.

14. The movement of charge in an electric field from one point ot another at a constant speed without the expenditure of work, by or against the field: (a) is impossible (b) can only occur along a field line (c) can only occur along an equipotiential (d) can only occur in a uniform field (e) none of these. Explain.

15. A neutron somehow picks up 10eV, which is equivalent to it increasing its: (a) charge by 10C (b) elecrical potential by 10V (c) energy by 16 X 10 -19J

(d) capacitance by 10 uF. Explain.

16. The electrostatic potential everywhere inside a hollow conductor is: (a) always zero (b) never positive (c) always a nonzero constant (d) constant, provided there are no enclosed isolated charges (e) none of these. Explain your choice.

17) The capacitance of a parallel-plate capacitor is: (a) independent of the plate seperation (b) dependent on the charge (c) dependent on the voltage (d) independent of the plate area (e) none of these. Explain your choice.

18) If the voltage across a capacitor is doubled, the amount of energy it can store: (a) doubles (b) is halved (c) is quadrupled (d) is unaffected (e) none of these. Explain your choice.

Problem:

1) A tiny positive charge of 50.0 uC is in a nonuniform electric field. An external force carries the charge from the original location, where the potential is 200.0 V, to a new location expending 100.0 mJ of work in the process. What is the potential of this new location?

2) Two charged parallel metal plates, inside the evacuated cathod-ray tube of a radar system, are seperated by 1.00 cm and have a potential difference of 25.0 V. What is the value of the electric field in the gap?

3) A proton is released from rest in a uniform electric field of 500 V/m. How fast will it be moving after traveling 40 cm in and parallel to the field?

4) A 10.0-cm-diameter solid gold sphere carries a charge of +0.100 uC. What is the potential 10.0 m away in the surrounding air?

5) A hydrogen atom consists of a proton around which circulates an elecron at an average distance of 0.053nm. Determine the potential at the distance due to the proton and find the potential energy of the electron in joules.

6) Two steel ball bearings 1.00 m apart in air carry charges of +40.0 uC and -20 uC. What is the potential at a point on the center-to-center line midway between them?

7) A +60.0-nC point-charge is at each of the vertices of an equilateral triagle. What is the potential at a point 25.0 cm from each of the charges?

8) A 48.0-uF capacitor, with an impregnated paper dielectric, is placed across the terminals of a 12-V battery. How much charge flows form the battery to the capacitor?

9) A 10.0-uF parallel-plate air capacitor is charged so it carries -200 uC on one plate and +200 uC on the other. What is the potential difference across it?

10) Two capacitors of 175 uF and 200 uF are wired in parallel. What's their equivalent capacitance?

11) Three 3.0-uF capacitors are arranged in series and put in a box so that only the first and last terminals jut out. What capacitance would be measured across those two terminals?

12) Imagine four 1.0-uF capacitors forming two groups, each consisting of two capacitors in parallel. These two pairs are then wired in series. What is the equivalent capacitance of all four?

13) Two capacitors (one 10.0 uF, the other 20.0 uF) are connected in series and the combination is placed across a 12.0 V battery. How much energy is stored in the system once it's fully charged?

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