Assignment 11. A positively charged wire is bent into a semicircle of radius , as shown in the figurebelow.The total charge on the semicircle is Q. However, the charge per unit length along thesemicircle is non-uniform and given by ? = ?o cos?.a) What is the relationship between ?o, R and Q?b) If a particle with a charge q is placed at the origin, what is the total force on theparticle? Show all your work including setting up and integrating any necessary integrals.2. Two thin, semi-infinite rods lie in the same plane. They make an angle of 45? with eachother and they are joined by another thin rod bent along an arc of a circle of radius R,with center at P. All the rods carry a uniform charge distribution of ? [C/m]. Find theelectric field at point P.3. The equipotentials for a potential landscape (on a 1 cm grid) are shown in the figure.The equipotential curves (the magenta circles) are marked at V = 0.25 V, 0.5V and thenfrom V = 1 V to V = 10 V in 1 V increments.Use the convention that red is the positive electrode (V = +10 V) and blue is ground (V =0 V).(a) Copy the above figure (in a separate graph) and sketch eight electric field lines on it(equally spaced around the inner conductor).(b) What, approximately, is the magnitude of the electric field at r = 1 cm, 2 cm, and 3cm, where r is measured from the center of the inner conductor? You should express thefield in V/cm. (HINT: The field is the local slope (derivative) of the potential. Also, ifyou choose to use a ruler realize that the above reproduction of this groups results is notthe same size as the original, where the grid size was 1 cm).(c) What is the relationship between the density of the equipotential lines, the density ofthe electric field lines, and the strength of the electric field?(d) Plot the field strength vs. 1/r2for the three points from part (a). If the field werecreated by a single point charge what shape should this sketch be? Is it?(e) Approximately how much charge was on the inner conductor when the group madetheir measurements?4. Consider two nested, spherical conducting shells. The first has inner radius a and outerradius b. The second has inner radius c and outer radius d.In the following four situations, determine the total charge on each of the faces of theconducting spheres (inner and outer for each), as well as the electric field and potentialeverywhere in space (as a function of distance r from the center of the spherical shells).In all cases the shells begin uncharged, and a charge is then instantly introducedsomewhere.a. Both shells are floating that is, their net charge will remain fixed. A positivecharge +Q is introduced into the center of the inner spherical shell. Take the zeroof potential to be at infinity.b. The inner shell is floating but the outer shell is grounded that is, it is fixed atV=0 and has whatever charge is necessary on it to maintain this potential. Anegative charge Q is introduced into the center of the inner spherical shell.c. The inner shell is grounded but the outer shell is floating. A positive charge +Q isintroduced into the center of the inner spherical shell.d. Finally, the outer shell is grounded and the inner shell is floating. This time thepositive charge +Q is introduced into the region in between the two shells. In thiscase the questions What is E(r)/V(r)? are not well defined in some regions ofspace. In the regions where these questions can be answered, answer them. In theregions where they cant be answered, explain why, and give as much informationabout the potential as possible (is it positive or negative, for example).; Assignment 11. A positively charged wire is bent into a semicircle of radius , as shown in the figurebelow.The total charge on the semicircle is Q. However, the charge per unit length along thesemicircle is non-uniform and given by ? = ?o cos?.a) What is the relationship between ?o, R and Q?b) If a particle with a charge q is placed at the origin, what is the total force on theparticle? Show all your work including setting up and integrating any necessary integrals.2. Two thin, semi-infinite rods lie in the same plane. They make an angle of 45? with eachother and they are joined by another thin rod bent along an arc of a circle of radius R,with center at P. All the rods carry a uniform charge distribution of ? [C/m]. Find theelectric field at point P.3. The equipotentials for a potential landscape (on a 1 cm grid) are shown in the figure.The equipotential curves (the magenta circles) are marked at V = 0.25 V, 0.5V and thenfrom V = 1 V to V = 10 V in 1 V increments.Use the convention that red is the positive electrode (V = +10 V) and blue is ground (V =0 V).(a) Copy the above figure (in a separate graph) and sketch eight electric field lines on it(equally spaced around the inner conductor).(b) What, approximately, is the magnitude of the electric field at r = 1 cm, 2 cm, and 3cm, where r is measured from the center of the inner conductor? You should express thefield in V/cm. (HINT: The field is the local slope (derivative) of the potential. Also, ifyou choose to use a ruler realize that the above reproduction of this groups results is notthe same size as the original, where the grid size was 1 cm).(c) What is the relationship between the density of the equipotential lines, the density ofthe electric field lines, and the strength of the electric field?(d) Plot the field strength vs. 1/r2for the three points from part (a). If the field werecreated by a single point charge what shape should this sketch be? Is it?(e) Approximately how much charge was on the inner conductor when the group madetheir measurements?4. Consider two nested, spherical conducting shells. The first has inner radius a and outerradius b. The second has inner radius c and outer radius d.In the following four situations, determine the total charge on each of the faces of theconducting spheres (inner and outer for each), as well as the electric field and potentialeverywhere in space (as a function of distance r from the center of the spherical shells).In all cases the shells begin uncharged, and a charge is then instantly introducedsomewhere.a. Both shells are floating that is, their net charge will remain fixed. A positivecharge +Q is introduced into the center of the inner spherical shell. Take the zeroof potential to be at infinity.b. The inner shell is floating but the outer shell is grounded that is, it is fixed atV=0 and has whatever charge is necessary on it to maintain this potential. Anegative charge Q is introduced into the center of the inner spherical shell.c. The inner shell is grounded but the outer shell is floating. A positive charge +Q isintroduced into the center of the inner spherical shell.d. Finally, the outer shell is grounded and the inner shell is floating. This time thepositive charge +Q is introduced into the region in between the two shells. In thiscase the questions What is E(r)/V(r)? are not well defined in some regions ofspace. In the regions where these questions can be answered, answer them. In theregions where they cant be answered, explain why, and give as much informationabout the potential as possible (is it positive or negative, for example).
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