Concepts & Calculations

Note: Each of these problems consists of Concept Questions followed by a related quantitative Problem. The Concept Questions involve little or no mathematics. They focus on the concepts with which the problems deal. Recognizing the concepts is the essential initial step in any problem-solving technique.

| 61 | Concept Questions The drawing shows a square, on two corners of which are fixed different positive charges. A third charge (negative) is brought to one of the empty corners. (a) At which corner, A or B, is the potential greater? (b) Is the electric potential energy of the third charge positive or negative? (c) At which corner, A or B, is the magnitude of the electric potential energy greater? Explain your answers.

Problem The length of a side of the square is . Find the electric potential energy of a charge placed at corner A and then at corner B. Compare your answers for consistency with your answers to the Concept Questions.

| 62 | Concept Questions Charges of −q and +2q are fixed in place, with a distance d between them. A dashed line is drawn through the negative charge, perpendicular to the line between the charges. On the dashed line, at a distance L from the negative charge, there is at least one spot where the total potential is zero. (a) At this spot, is the magnitude of the potential from the positive charge greater than, less than, or equal to the magnitude of the potential from the negative charge? (b) Is the distance from the positive charge to the zero-potential spot greater than, less than, or equal to L? (c) How many spots on the dashed line are there where the total potential is zero? Account for your answers.
Problem The distance between the charges is . Find L.

| 63 | Concept Questions An electron and a proton, starting from rest, are accelerated through an electric potential difference of the same magnitude. In the process, the electron acquires a speed ve, while the proton acquires a speed vp. (a) As each particle accelerates from rest, it gains kinetic energy. Does it gain or lose electric potential energy? (b) Does the electron gain more, less, or the same amount of kinetic energy as the proton does? (c) Is ve greater than, less than, or equal to vp? Justify your answers.
Problem Find the ratio ve/vp. Verify that your answer is consistent with your answers to the Concept Questions.

| 64 | Concept Questions A positive point charge is surrounded by an equipotential surface A, which has a radius of rA. A positive test charge moves from surface A to another equipotential surface B, which has a radius of rB. In the process, the electric force does negative work. (a) Does the electric force acting on the test charge have the same or opposite direction as the displacement of the test charge? (b) Is rB greater than or less than rA? Explain your answers.
Problem The positive point charge is , and the test charge is . The work done as the test charge moves from surface A to surface B is . The radius of surface A is . Find rB. Check to see that your answer is consistent with your answers to the Concept Questions.

| 65 | Concept Questions Two capacitors have the same plate separation. However, one has square plates, while the other has circular plates. The square plates are a length L on each side, and the diameter of the circular plates is L. (a) If the same dielectric material were between the plates in each capacitor, which one would have the greater capacitance? (b) By putting different dielectric materials between the capacitor plates, we can make the two capacitors have the same capacitance. Which capacitor should contain the dielectric material with the greater dielectric constant? Give your reasoning in each case.
Problem The capacitors have the same capacitance because they contain different dielectric materials. The dielectric constant of the material between the square plates has a value of . What is the dielectric constant of the material between the circular plates? Be sure that your answer is consistent with your answers to the Concept Questions.

| 66 | Concept Questions Capacitor A and capacitor B both have the same voltage across their plates. However, the energy of capacitor A can melt m kilograms of ice at 0 °C, while the energy of capacitor B can boil away the same amount of water at 100 °C. (a) Which requires more energy, melting the ice or boiling the water? (b) Which capacitor has the greater capacitance? Explain your answers.
Problem The capacitance of capacitor A is . What is the capacitance of capacitor B? Be sure that your answer is consistent with your answers to the Concept Questions.

| 67 | Concept Questions During a lightning flash, a potential difference exists between a cloud and the ground. As a result, negative electric charge is transferred from the ground to the cloud. (a) As the charge moves from the ground to the cloud, work is done on the charge by the electric force. How is this work related to the potential difference and the charge q? (b) If this work could be used to accelerate an automobile of mass m from rest, what would be the automobile’s final speed? Express your answer in terms of the potential difference, the charge, and the mass of the automobile. (c) If this work could all be converted into heat, what mass m of water could be heated so that its temperature increases by ? Write your answer in terms of the potential difference, the charge, the mass, the specific heat capacity c of water, and .
Problem Suppose a potential difference of exists between the cloud and the ground, and of charge is transferred from the ground to the cloud. (a) How much work is done on the charge by the electric force? (b) If the work done by the electric force were used to accelerate a 1100-kg automobile from rest, what would be its final speed? (c) If the work done by the electric force were converted into heat, how many kilograms of water at 0 °C could be heated to 100 °C?

| 68 | Concept Questions An electron is released at the negative plate of a parallel plate capacitor and accelerates to the positive plate (see the drawing), (a) As the electron gains kinetic energy, does its electric potential energy increase or decrease? Why? (b) The difference in the electron’s electric potential energy between the positive and negative plates is . How is this difference related to the charge on the electron (−e) and to the difference in the electric potential between the plates? (c) How is the potential difference related to the electric field within the capacitor and the displacement of the positive plate relative to the negative plate?

Problem The plates of a parallel plate capacitor are separated by a distance of 1.2 cm, and the electric field within the capacitor has a magnitude of . An electron starts from rest at the negative plate and accelerates to the positive plate. What is the kinetic energy of the electron just as the electron reaches the positive plate?

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