2) Consider a point located at a distance a above the edge of a "semi-infinite" line of charge.. The line has a uniform charge density A throughout its length. Here, A is in units of Coulombs per meter and has a positive value: therefore the line itself has a net positive charge. Also, the thickness of the line itself can be ignored. For this problem, the point of interest can be envisioned as the origin of an (r,y) coordinate system, with positive r pointed toward the right and positive y pointed upward. The line is oriented such that it is parallel to the r-axis and the position of the point (as well as the near edge of the line) is r=0. The line of charge extends from r=0 to r=+o. Using this information, calculate the electric field E at the location of the point. Follow the steps below to achieve your final answer. a) Recall that a differential element dE of the electric field in terms of the differential charge element dq and the distance r may be expressed as 1 dq dE In this particular problem, what is dq as a function of A and a differential line element dr along the line? Here, r is the distance from the point to the differential line element dr. b) At the point of interest, will there be a non-zero component of dE in the r-direction (call this component dEx)? Will there be a non-zero component of dE in the y-direction (call this component dEy)? Note that through these definitions dE = dEx + dEy. c) Express dEx and dEy in terms of dE and the appropriate trigonometric functions with 0 as the argument. Here, 0 is defined to be the angle between the leg connecting the point and the near edge of the line (this leg has a length a) and a leg connecting the point and the position dr along the line of charge (this leg has a length r). d) From the definition of a given in Part (c), what is cos 0 in terms of a and r and what is sin 0 in terms of r and r? e) Based on your answers to Parts (a) and (d), determine Ex and Ey – that is, the integrations of dEx and dEy. Over which variable do you perform the integrations for dEx and dEy? What then is the direction of the electric field at the point of interest and therefore in which direction with the positive test charge be accelerated? Explain your answer.
2) Consider a point located at a distance a above the edge of a "semi-infinite" line of charge.. The line has a uniform charge density A throughout its length. Here, A is in units of Coulombs per meter and has a positive value: therefore the line itself has a net positive charge. Also, the thickness of the line itself can be ignored. For this problem, the point of interest can be envisioned as the origin of an (r,y) coordinate system, with positive r pointed toward the right and positive y pointed upward. The line is oriented such that it is parallel to the r-axis and the position of the point (as well as the near edge of the line) is r=0. The line of charge extends from r=0 to r=+o. Using this information, calculate the electric field E at the location of the point. Follow the steps below to achieve your final answer. a) Recall that a differential element dE of the electric field in terms of the differential charge element dq and the distance r may be expressed as 1 dq dE In this particular problem, what is dq as a function of A and a differential line element dr along the line? Here, r is the distance from the point to the differential line element dr. b) At the point of interest, will there be a non-zero component of dE in the r-direction (call this component dEx)? Will there be a non-zero component of dE in the y-direction (call this component dEy)? Note that through these definitions dE = dEx + dEy. c) Express dEx and dEy in terms of dE and the appropriate trigonometric functions with 0 as the argument. Here, 0 is defined to be the angle between the leg connecting the point and the near edge of the line (this leg has a length a) and a leg connecting the point and the position dr along the line of charge (this leg has a length r). d) From the definition of a given in Part (c), what is cos 0 in terms of a and r and what is sin 0 in terms of r and r? e) Based on your answers to Parts (a) and (d), determine Ex and Ey – that is, the integrations of dEx and dEy. Over which variable do you perform the integrations for dEx and dEy? What then is the direction of the electric field at the point of interest and therefore in which direction with the positive test charge be accelerated? Explain your answer.
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Transcribed Image Text:2) Consider a point located at a distance a above the edge of a "semi-infinite" line of charge.. The
line has a uniform charge density A throughout its length. Here, A is in units of Coulombs per meter
and has a positive value: therefore the line itself has a net positive charge. Also, the thickness of the
line itself can be ignored. For this problem, the point of interest can be envisioned as the origin of
an (r,y) coordinate system, with positive r pointed toward the right and positive y pointed upward.
The line is oriented such that it is parallel to the r-axis and the position of the point (as well as the
near edge of the line) is r=0. The line of charge extends from r=0 to r=+o. Using this information,
calculate the electric field E at the location of the point. Follow the steps below to achieve your final
answer.
a) Recall that a differential element dE of the electric field in terms of the differential charge element
dq and the distance r may be expressed as
1 dq
dE
In this particular problem, what is dq as a function of A and a differential line element dr along the
line? Here, r is the distance from the point to the differential line element dr.
b) At the point of interest, will there be a non-zero component of dE in the r-direction (call this
component dEx)? Will there be a non-zero component of dE in the y-direction (call this component
dEy)? Note that through these definitions dE = dEx + dEy.
c) Express dEx and dEy in terms of dE and the appropriate trigonometric functions with 0 as the
argument. Here, 0 is defined to be the angle between the leg connecting the point and the near edge
of the line (this leg has a length a) and a leg connecting the point and the position dr along the line
of charge (this leg has a length r).
d) From the definition of a given in Part (c), what is cos 0 in terms of a and r and what is sin 0 in
terms of r and r?
e) Based on your answers to Parts (a) and (d), determine Ex and Ey – that is, the integrations of
dEx and dEy. Over which variable do you perform the integrations for dEx and dEy? What then
is the direction of the electric field at the point of interest and therefore in which direction with the
positive test charge be accelerated? Explain your answer.
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