Approximations for Spherical and Cylindrical Capacitor
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We associate a certain number of points with each exercise.
When you click an exercise into a collection, this number will be taken as points for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit the number of points for the exercise in the collection independently, without any effect on "points by default" as represented by the number here.
That being said... How many "default points" should you associate with an exercise upon creation?
As with difficulty, there is no straight forward and generally accepted way.
But as a guideline, we tend to give as many points by default as there are mathematical steps to do in the exercise.
Again, very vague... But the number should kind of represent the "work" required.
When you click an exercise into a collection, this number will be taken as points for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit the number of points for the exercise in the collection independently, without any effect on "points by default" as represented by the number here.
That being said... How many "default points" should you associate with an exercise upon creation?
As with difficulty, there is no straight forward and generally accepted way.
But as a guideline, we tend to give as many points by default as there are mathematical steps to do in the exercise.
Again, very vague... But the number should kind of represent the "work" required.
About difficulty...
We associate a certain difficulty with each exercise.
When you click an exercise into a collection, this number will be taken as difficulty for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit its difficulty in the collection independently, without any effect on the "difficulty by default" here.
Why we use chess pieces? Well... we like chess, we like playing around with \(\LaTeX\)-fonts, we wanted symbols that need less space than six stars in a table-column... But in your layouts, you are of course free to indicate the difficulty of the exercise the way you want.
That being said... How "difficult" is an exercise? It depends on many factors, like what was being taught etc.
In physics exercises, we try to follow this pattern:
Level 1 - One formula (one you would find in a reference book) is enough to solve the exercise. Example exercise
Level 2 - Two formulas are needed, it's possible to compute an "in-between" solution, i.e. no algebraic equation needed. Example exercise
Level 3 - "Chain-computations" like on level 2, but 3+ calculations. Still, no equations, i.e. you are not forced to solve it in an algebraic manner. Example exercise
Level 4 - Exercise needs to be solved by algebraic equations, not possible to calculate numerical "in-between" results. Example exercise
Level 5 -
Level 6 -
When you click an exercise into a collection, this number will be taken as difficulty for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit its difficulty in the collection independently, without any effect on the "difficulty by default" here.
Why we use chess pieces? Well... we like chess, we like playing around with \(\LaTeX\)-fonts, we wanted symbols that need less space than six stars in a table-column... But in your layouts, you are of course free to indicate the difficulty of the exercise the way you want.
That being said... How "difficult" is an exercise? It depends on many factors, like what was being taught etc.
In physics exercises, we try to follow this pattern:
Level 1 - One formula (one you would find in a reference book) is enough to solve the exercise. Example exercise
Level 2 - Two formulas are needed, it's possible to compute an "in-between" solution, i.e. no algebraic equation needed. Example exercise
Level 3 - "Chain-computations" like on level 2, but 3+ calculations. Still, no equations, i.e. you are not forced to solve it in an algebraic manner. Example exercise
Level 4 - Exercise needs to be solved by algebraic equations, not possible to calculate numerical "in-between" results. Example exercise
Level 5 -
Level 6 -
Question
Solution
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Exercise:
The distance between the conductors of a capacitor is often small compared to the area of the surfaces. In these situations the capacitance of a parallel plate capacitor is a good approximation. abcliste abc Show that the capacitance of a spherical capacitor with a small distance between the spherical shells can be approximated by a parallel plate capacitor with the same distance and an area that corresponds to the surface area. abc Show that the capacitance of a cylindrical capacitor with a small distance between the cylindrical surfaces can be approximated by a parallel plate capacitor with the same distance and an area that corresponds to the mantle area. abcliste
Solution:
abcliste abc The radii of the concentric shells are r and r+d. It follows for the capacitance C pivarepsilon_leftfracr-fracr+dright^- pivarepsilon_leftfracr+d-rrr+dright^- pivarepsilon_fracrr+dd For d ll r the distance between shells can be neglected compared to the radius i.e. r+dapprox r. It follows that C &approx pivarepsilon_fracr^d varepsilon_fracpi r^d varepsilon_fracAd with Api r^ i.e. the surface of a spherical shell with radius r. The graph shows the exact capacitance and the approximation as a function of the ratio d/r and the ratio of the two quantities right axis. center includegraphicswidthtextwidth#image_path:graph-spherical-capacitor# center abc The radii of the concentric cylinders are r and r+d. It follows for the capacitance C fracpivarepsilon_ Llnfracr+dr fracpivarepsilon_ Lln+d/r For small values of x the logarithm of +x can be approximated by x. This leads to C &approx fracpivarepsilon_ Ld/r varepsilon_fracpi r Ld varepsilon_fracAd with Api r L i.e. the mantle area of a cylinder with length L and radius r. The graph shows the exact capacitance and the approximation as a function of the ratio d/r and the ratio of the two quantities right axis. center includegraphicswidthtextwidth#image_path:graph-cylindrical-capacitor# center abcliste
The distance between the conductors of a capacitor is often small compared to the area of the surfaces. In these situations the capacitance of a parallel plate capacitor is a good approximation. abcliste abc Show that the capacitance of a spherical capacitor with a small distance between the spherical shells can be approximated by a parallel plate capacitor with the same distance and an area that corresponds to the surface area. abc Show that the capacitance of a cylindrical capacitor with a small distance between the cylindrical surfaces can be approximated by a parallel plate capacitor with the same distance and an area that corresponds to the mantle area. abcliste
Solution:
abcliste abc The radii of the concentric shells are r and r+d. It follows for the capacitance C pivarepsilon_leftfracr-fracr+dright^- pivarepsilon_leftfracr+d-rrr+dright^- pivarepsilon_fracrr+dd For d ll r the distance between shells can be neglected compared to the radius i.e. r+dapprox r. It follows that C &approx pivarepsilon_fracr^d varepsilon_fracpi r^d varepsilon_fracAd with Api r^ i.e. the surface of a spherical shell with radius r. The graph shows the exact capacitance and the approximation as a function of the ratio d/r and the ratio of the two quantities right axis. center includegraphicswidthtextwidth#image_path:graph-spherical-capacitor# center abc The radii of the concentric cylinders are r and r+d. It follows for the capacitance C fracpivarepsilon_ Llnfracr+dr fracpivarepsilon_ Lln+d/r For small values of x the logarithm of +x can be approximated by x. This leads to C &approx fracpivarepsilon_ Ld/r varepsilon_fracpi r Ld varepsilon_fracAd with Api r L i.e. the mantle area of a cylinder with length L and radius r. The graph shows the exact capacitance and the approximation as a function of the ratio d/r and the ratio of the two quantities right axis. center includegraphicswidthtextwidth#image_path:graph-cylindrical-capacitor# center abcliste
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Exercise:
The distance between the conductors of a capacitor is often small compared to the area of the surfaces. In these situations the capacitance of a parallel plate capacitor is a good approximation. abcliste abc Show that the capacitance of a spherical capacitor with a small distance between the spherical shells can be approximated by a parallel plate capacitor with the same distance and an area that corresponds to the surface area. abc Show that the capacitance of a cylindrical capacitor with a small distance between the cylindrical surfaces can be approximated by a parallel plate capacitor with the same distance and an area that corresponds to the mantle area. abcliste
Solution:
abcliste abc The radii of the concentric shells are r and r+d. It follows for the capacitance C pivarepsilon_leftfracr-fracr+dright^- pivarepsilon_leftfracr+d-rrr+dright^- pivarepsilon_fracrr+dd For d ll r the distance between shells can be neglected compared to the radius i.e. r+dapprox r. It follows that C &approx pivarepsilon_fracr^d varepsilon_fracpi r^d varepsilon_fracAd with Api r^ i.e. the surface of a spherical shell with radius r. The graph shows the exact capacitance and the approximation as a function of the ratio d/r and the ratio of the two quantities right axis. center includegraphicswidthtextwidth#image_path:graph-spherical-capacitor# center abc The radii of the concentric cylinders are r and r+d. It follows for the capacitance C fracpivarepsilon_ Llnfracr+dr fracpivarepsilon_ Lln+d/r For small values of x the logarithm of +x can be approximated by x. This leads to C &approx fracpivarepsilon_ Ld/r varepsilon_fracpi r Ld varepsilon_fracAd with Api r L i.e. the mantle area of a cylinder with length L and radius r. The graph shows the exact capacitance and the approximation as a function of the ratio d/r and the ratio of the two quantities right axis. center includegraphicswidthtextwidth#image_path:graph-cylindrical-capacitor# center abcliste
The distance between the conductors of a capacitor is often small compared to the area of the surfaces. In these situations the capacitance of a parallel plate capacitor is a good approximation. abcliste abc Show that the capacitance of a spherical capacitor with a small distance between the spherical shells can be approximated by a parallel plate capacitor with the same distance and an area that corresponds to the surface area. abc Show that the capacitance of a cylindrical capacitor with a small distance between the cylindrical surfaces can be approximated by a parallel plate capacitor with the same distance and an area that corresponds to the mantle area. abcliste
Solution:
abcliste abc The radii of the concentric shells are r and r+d. It follows for the capacitance C pivarepsilon_leftfracr-fracr+dright^- pivarepsilon_leftfracr+d-rrr+dright^- pivarepsilon_fracrr+dd For d ll r the distance between shells can be neglected compared to the radius i.e. r+dapprox r. It follows that C &approx pivarepsilon_fracr^d varepsilon_fracpi r^d varepsilon_fracAd with Api r^ i.e. the surface of a spherical shell with radius r. The graph shows the exact capacitance and the approximation as a function of the ratio d/r and the ratio of the two quantities right axis. center includegraphicswidthtextwidth#image_path:graph-spherical-capacitor# center abc The radii of the concentric cylinders are r and r+d. It follows for the capacitance C fracpivarepsilon_ Llnfracr+dr fracpivarepsilon_ Lln+d/r For small values of x the logarithm of +x can be approximated by x. This leads to C &approx fracpivarepsilon_ Ld/r varepsilon_fracpi r Ld varepsilon_fracAd with Api r L i.e. the mantle area of a cylinder with length L and radius r. The graph shows the exact capacitance and the approximation as a function of the ratio d/r and the ratio of the two quantities right axis. center includegraphicswidthtextwidth#image_path:graph-cylindrical-capacitor# center abcliste
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