An annulus, shown on the right, has inner and outer radii R₁ and R₂ and a uniform surface charge density of o (in units of C/m²). a) What is the total charge Q on the annulus? b) For points along the z-axis, show that the y- and z-components of the electric field are zero and that the z-component is given by E₂(x) = OI √2²+ +R²² √√7²²+R²₂ [Hint: If you aren't sure where to start, first determine the separation vector pointing from an arbitrary point on the annulus to a point on the z-axis. This'll be a function of r as well as two 'dummy' variables that parameterize the charge configuration. Because of the shape of the annulus, I suggest using polar coordinates (making the dummy variables that you'll integrate over r and 0) in the yz-plane to parameterize it, with area element dA=rdfdr. To find the total electric field at z, you'll need to integrate the field produced by each point on the annulus over its full area to get a final quantity dependent only on z.] c) For [2] << 1 (as in very small) we can use the first two terms of the Taylor approximation (1+z)ª ≈ 1+az. By inserting appropriate values for a and z, show that for z>> R₂, the electric field reduces to the one produced by a point charge Q, that is, with the same total charge computed in part a). d) Using the same approximation, now with a different value for z, show that for r << R₁, the magnitude of the electric field is lincar as a function of z. If we place a charge q with mass m at the center of the annulus, constrained to the z-axis, and give it a little nudge in the z-direction, what will its frequency of oscillation be?
An annulus, shown on the right, has inner and outer radii R₁ and R₂ and a uniform surface charge density of o (in units of C/m²). a) What is the total charge Q on the annulus? b) For points along the z-axis, show that the y- and z-components of the electric field are zero and that the z-component is given by E₂(x) = OI √2²+ +R²² √√7²²+R²₂ [Hint: If you aren't sure where to start, first determine the separation vector pointing from an arbitrary point on the annulus to a point on the z-axis. This'll be a function of r as well as two 'dummy' variables that parameterize the charge configuration. Because of the shape of the annulus, I suggest using polar coordinates (making the dummy variables that you'll integrate over r and 0) in the yz-plane to parameterize it, with area element dA=rdfdr. To find the total electric field at z, you'll need to integrate the field produced by each point on the annulus over its full area to get a final quantity dependent only on z.] c) For [2] << 1 (as in very small) we can use the first two terms of the Taylor approximation (1+z)ª ≈ 1+az. By inserting appropriate values for a and z, show that for z>> R₂, the electric field reduces to the one produced by a point charge Q, that is, with the same total charge computed in part a). d) Using the same approximation, now with a different value for z, show that for r << R₁, the magnitude of the electric field is lincar as a function of z. If we place a charge q with mass m at the center of the annulus, constrained to the z-axis, and give it a little nudge in the z-direction, what will its frequency of oscillation be?
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![An annulus, shown on the right, has inner and outer radii R₁ and R₂ and a
uniform surface charge density of o (in units of C/m²).
a) What is the total charge Q on the annulus?
b) For points along the z-axis, show that the y- and z-components of the
electric field are zero and that the 2-component is given by
E₂(x) =
+R²
R²₂
[Hint: If you aren't sure where to start, first determine the separation vector
pointing from an arbitrary point on the annulus to a point on the z-axis.
This'll be a function of r as well as two 'dummy' variables that parameterize the charge configuration.
Because of the shape of the annulus, I suggest using polar coordinates (making the dummy variables that
you'll integrate over r and 0) in the yz-plane to parameterize it, with area element dA= rdfdr. To find
the total electric field at r, you'll need to integrate the field produced by each point on the annulus over
its full area to get a final quantity dependent only on z.]
c) For [2] << 1 (as in very small) we can use the first two terms of the Taylor approximation (1+z)ª
1+az. By inserting appropriate values for a and z, show that for z>> R₂, the electric field reduces to
the one produced by a point charge Q, that is, with the same total charge computed in part a).
d) Using the same approximation, now with a different value for z, show that for z << R₁, the magnitude
of the electric field is lincar as a function of z. If we place a charge q with mass m at the center of the
annulus, constrained to the z-axis, and give it a little nudge in the 2-direction, what will its frequency of
oscillation be?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F474a784e-9f95-4813-81d7-a60d3fc34f86%2F4ac72957-fedf-430f-b4a6-d588ddb39a09%2Flhhvpkb_processed.png&w=3840&q=75)
Transcribed Image Text:An annulus, shown on the right, has inner and outer radii R₁ and R₂ and a
uniform surface charge density of o (in units of C/m²).
a) What is the total charge Q on the annulus?
b) For points along the z-axis, show that the y- and z-components of the
electric field are zero and that the 2-component is given by
E₂(x) =
+R²
R²₂
[Hint: If you aren't sure where to start, first determine the separation vector
pointing from an arbitrary point on the annulus to a point on the z-axis.
This'll be a function of r as well as two 'dummy' variables that parameterize the charge configuration.
Because of the shape of the annulus, I suggest using polar coordinates (making the dummy variables that
you'll integrate over r and 0) in the yz-plane to parameterize it, with area element dA= rdfdr. To find
the total electric field at r, you'll need to integrate the field produced by each point on the annulus over
its full area to get a final quantity dependent only on z.]
c) For [2] << 1 (as in very small) we can use the first two terms of the Taylor approximation (1+z)ª
1+az. By inserting appropriate values for a and z, show that for z>> R₂, the electric field reduces to
the one produced by a point charge Q, that is, with the same total charge computed in part a).
d) Using the same approximation, now with a different value for z, show that for z << R₁, the magnitude
of the electric field is lincar as a function of z. If we place a charge q with mass m at the center of the
annulus, constrained to the z-axis, and give it a little nudge in the 2-direction, what will its frequency of
oscillation be?
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