HW11Due: 11:59pm on Monday, September 28, 2009 Note: To understand how points are awarded, read your instructor's Grading Policy. [Return to Standard Assignment View] In Parts A-F of "Electric Force and Potential: Spherical Symmetry", the point charge is positive and the zero of potential is understood to be an infinite distance from the point charge. GBA Electric Force and Potential: Spherical Symmetry Learning Goal: To understand the electric potential and electric field of a point charge in three dimensions Consider a positive point charge , located at the origin of three-dimensional space. Throughout this problem, use in place of . Part A Due to symmetry, the electric field of a point charge at the origin must point _____ from the origin. Answer in one word. ANSWER: away Correct Part B Find , the magnitude of the electric field at distance from the point charge . Express your answer in terms of , , and . ANSWER: = Correct Part C Find , the electric potential at distance from the point charge . Express your answer in terms of , , and . ANSWER: = Correct Part D Which of the following is the correct relationship between the magnitude of a radial electric field electric potential and its associated ? More than one answer may be correct for the particular case of a point charge at the origin, but you should choose the correct general relationship. ANSWER: Correct Now consider the figure, which shows several functions of the variable . Part E Which curve could indicate the magnitude of the electric field due to a charge Hint E.1 How to approach the problem Hint not displayed located at the origin ( )? 2 Conceptualizing electric potential Hint not displayed Rank the locations from largest to smallest potential. ANSWER: .1 Definition of electric potential Hint not displayed Hint A.ANSWER: A B C D E F Correct Part F Which curve could indicate the electric potential due to a positive charge Hint F. You will be asked about the electric potential at the different points (A through F). Rank positive electric potentials as larger than negative electric potentials. Hint A. ANSWER: higher to lower lower to higher Correct HINT: In both of these ranking tasks. and . To rank items as equivalent. the electric field points from regions of ______ electric potential. Part A Rank the locations A to F on the basis of the electric potential at each point. overlap them. at various distances from the two point charges. GBA Electric Potential Ranking Task In the figurethere are two point charges.1 How to approach the problem Hint not displayed ANSWER: A B C D E F Correct located at the origin ( )? Part G Which curve could indicate the electric potential due to a negative charge ANSWER: A B C D E F Correct located at the origin ( )? Part H For either a positive or a negative charge. at least two of the six items should be ranked equally. You do this by putting those two items in the same column in the ranking box. labeled A through F. There are also six positions. ANSWER: -300 V Correct . due to the first two charges. and "Potential of a Finite Rod". GBA Potential of a Charged Ring A ring with radius and a uniformly distributed total charge lies in the xy plane. and a charge . ANSWER: View Correct Think carefully about the sign of your answer for Part (c) of Problem 23. you MAY type your answers in terms of k. Part C If the third charge moves from the point done on it by the field of the first two charges. Do not believe any MP error responses that read "The correct answer does not depend on the variable: k".31×10 −6 J Correct . due to the first two charges. Let the potential be zero far from the is placed at the origin of an xy-coordinate system. Rank the largestmagnitude positive change (increase in electric potential) as largest and the largest-magnitude negative change (decrease in electric potential) as smallest. You will be asked to rank changes in the electric potential along paths between pairs of points. Due to a bug in these three problems. "Potential of a Charged Annulus". to the point .2 Determine the algebraic sign of the change in potential Hint not displayed Hint A.1 Change in electric potential Hint not displayed Hint A.3 Conceptualizing changes in electric potential Hint not displayed Rank from largest to smallest. A third charge is now placed at the point .View Correct Change in Electric Potential Ranking Task In the diagram below. To rank items as equivalent. . GBA Problem 23. Make sure that you understand the physical reason for this sign. ANSWER: −4. you should regard it as "Try Again". If you get this response. Part A Using the diagram to the left. overlap them. . calculate the work NOTE: In "Potential of a Charged Ring". there are two charges of and and six points (a through f) at various distances from the two charges.66 A charge positive x-axis at Part A Calculate the potential at the point charges. MP sometimes gives this response when the error is something else entirely. Part B Calculate the potential at the point ANSWER: 419 V Correct . is placed on the . centered at the origin.66. Hint A. rank each of the given paths on the basis of the change in electric potential. . ANSWER: | |= Correct Notice that while the potential is a strictly decreasing function of . and or .1 due to the ring on the z axis as a function of ? How to approach the problem The formula for the electric potential produced by a static charge distribution involves the amount of charge and the distance from the charge to the position where the potential is measured. these vector cancellations.1 on the z axis as a function of . . per unit length of the ring For the "Potential of a Charged Annulus". This enables you to calculate the electric potential simply. for ? Determine the direction of the field By symmetry. Why does the electric field exhibit such a behavior? Though the contribution to the electric field from each point on the ring strictly decreases as a function of . All points on the ring are equidistant from a given point on the z axis.Part A What is the potential Hint A. even though all the individual on account of 's point in (almost) the as . because the contribution to the electric field. is the magnitude of the charge. the electric field has only one Cartesian component. From our work with electric field integration. I do not recommend following the MP hints. the electric field first increases till starts to decrease. . . Hint A. and or . with its center at the origin of the coordinate axes. you will need only to know the formula for potential of a point charge: . Express your answer in terms of . and the permittivity of free space. where is the potential at distance from the point charge. . without doing an integral. sits in the xy plane.2 The potential due to a point charge is If you incorporate the symmetry of the problem. GBA Potential of a Charged Annulus An annular ring with a uniform surface charge density The annulus has an inner radius and outer radius .2 The relationship between electric field and potential Hint not displayed Express your answer in terms of some or all of the quantities . and then same direction there. the vector cancellation from points on opposite sides of the ring becomes very strong for small . ANSWER: = Correct Part B What is the magnitude of the electric field Hint B. you should know how to write an integral over an annulus. and in this case the integral is simpler to write since potential is a scalar rather than a vector. On the other hand . In what direction does the electric field point? ANSWER: Correct Hint B. and radius . and radius ? Express your answer in terms of ANSWER: = Correct Now do the integral over . This will put the integral . calculate the electric potential distance Hint B. .4 A formula for the integral Hint not displayed Express your answer in terms of some or all of the variables . However. Don't forget that is a function of . How to exploit the angular symmetry of the problem at a point on the axis of the annulus can be written as . and those on the horizontal line C.Part A If you can find symmetries in a physical situation. The integral over in the form is easy and should be done first. Use .1 Definition of the potential due to a point charge Hint not displayed ANSWER: points on line A points on circle B points on line C Correct Part B By exploiting the above symmetry. . Hint B. ANSWER: = Correct It is interestering to note that the potential at any point on the axis of a disk of radius can be obtained from the . Which set of points makes the same contribution toward the potential calculated at any point along the axis of the annulus? Hint A.2 What is is the area of a thin annular slice of thickness Find the area of an annular slice . or otherwise. at a point on the axis of the annulus a The total potential where is the distance from a point on the annulus to the point at which the potential is to be determined. In this part you will find a symmetry in the annular ring before calculating the potential along the axis through the ring's center in Part B. as shown in the figure.3 Doing the integral Hint not displayed Hint B. it is more convenient to write this integral in terms of polar coordinates: . where Hint B. on account of the angular symmetry of this problem. You will need to use a variable substitution. those on circle B. you can often greatly simplify your calculations. the area of a thin annular slice of thickness and .1 from its center. since the integrand has no dependence on . and . Consider three sets of points: points lying on the vertical line A. this is the case. because it means that from far away. which intuitively seems reasonable. Clearly. one obtains . Part B What is . one with charge density and the other with charge density and radius . this means that the logarithm can be further approximated as . Conversely. Thus one endpoint is located at zero at an infinite distance away from the rod. .3 A helpful integral Hint not displayed Express your answer in terms of ANSWER: = Correct .2 Find the electric potential of a section of the rod Hint not displayed Hint A. just like the potential due to a point charge. so the net charge distribution would be just like that of the annulus. the annulus can be thought of as the superposition of two disks. Doing so. by adding the potentials due to these two disks. This is what we expect. Indeed. If . For . of which one is subtracted from the other. located at distance from one end of the rod (on the x axis)? . the electric potential at point A (see the figure). using the formula above. Throughout this problem. Potential of a Finite Rod A finite rod of length has total charge . . However. and the expression for potential reduces to . the potential due to the charged rod looks like that due to a point charge. . For this problem. . you would recover the potential of the annulus. Try some values or check that the derivative of is indeed negative. the potential increases with increasing charge densities. You can also check that the above expression actually reduces to the potential due to a point charge for . the opposite charge densities cancel out. above the midpoint of the rod on the y axis? located a distance Hint A. distributed uniformly along its length. on account of the negative second term in the expressions. and . Moreover. In the region from the center to and radius . this answer can be approximated as . Part A What is . and the potential should drop off. the disk looks like a point. The rod lies on the x -axis and is centered at . Define the electric potential to be in place of the expression the origin.1 How to approach the problem Hint not displayed Hint A. if the distance increases. It is also instructive to look at the general behavior of these potentials as a function of the parameters. as well as with increasing areas (if the charge density is held constant).expression above by setting and . the electric potential at point . since appears in both terms. and the other is located at . you may use the constant . it is not clear whether the potential should grow. If you are far from the disk. . ANSWER: = Correct . You received 49.3%. . the potential due to the charged rod looks like that due to a point charge.1 How to approach the problem Hint not displayed Hint B.2 Find the distance from point B to a segment of the rod Hint not displayed Give your answer in terms of . Score Summary: Your score on this assignment is 98. the logarithm approaches . for to . and .17 out of a possible total of 50 points. As before. because it means that from far away. This is what we expect. This result can be written as . Thus.Hint B. for . . in which case the result reduces .