Advanced Double Capacitor
About points...
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
Short
Video
\(\LaTeX\)
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Exercise:
A double capacitor consists of two fixed outer plates and two connected inner plates. The inner plates can be moved from left to right thereby changing the effective overlapping plate area of the two parallel plate capacitors with the same plate distance. When the area of the upper capacitor increases the area of the lower one decreases and vice versa see figure. abcliste abc Find an expression for the series capacitance of the two capacitors as a function of the position x. abc Determine the position x_ for the maximum capacitance. abcliste center includegraphicswidth.textwidth#image_path:advanced-doublcapacitor# center
Solution:
abcliste abc With a maximum plate area A and a distance d between the pairs of plates we can write for the two capacitances: C_ varepsilon_fracx Ad C_ varepsilon_frac-x Ad with xin . For the series capacitance we find sscCs leftfracC_+fracC_right^- leftfracdvarepsilon_ x A+fracdvarepsilon_ -x Aright^- varepsilon_fracAdleftfracx+frac-xright^- varepsilon_fracAdleftfrac-x+xx-xright^- varepsilon_fracAd x-x resultvarepsilon_fracAd x-x^ abc In order to find the maximum of the capacitance we can derive the expression from a with respect to the position x: fracmathrmdsscCsmathrmdx varepsilon_fracAd - x At an extremal po xx_ the derivative is equal to zero. This leads to - x_ x_ resultfrac The result can easily be verified with a graph of fxx-x^ in the range : center includegraphicswidth.textwidth#image_path:graph-advanced-doublcapacitor# center abcliste
A double capacitor consists of two fixed outer plates and two connected inner plates. The inner plates can be moved from left to right thereby changing the effective overlapping plate area of the two parallel plate capacitors with the same plate distance. When the area of the upper capacitor increases the area of the lower one decreases and vice versa see figure. abcliste abc Find an expression for the series capacitance of the two capacitors as a function of the position x. abc Determine the position x_ for the maximum capacitance. abcliste center includegraphicswidth.textwidth#image_path:advanced-doublcapacitor# center
Solution:
abcliste abc With a maximum plate area A and a distance d between the pairs of plates we can write for the two capacitances: C_ varepsilon_fracx Ad C_ varepsilon_frac-x Ad with xin . For the series capacitance we find sscCs leftfracC_+fracC_right^- leftfracdvarepsilon_ x A+fracdvarepsilon_ -x Aright^- varepsilon_fracAdleftfracx+frac-xright^- varepsilon_fracAdleftfrac-x+xx-xright^- varepsilon_fracAd x-x resultvarepsilon_fracAd x-x^ abc In order to find the maximum of the capacitance we can derive the expression from a with respect to the position x: fracmathrmdsscCsmathrmdx varepsilon_fracAd - x At an extremal po xx_ the derivative is equal to zero. This leads to - x_ x_ resultfrac The result can easily be verified with a graph of fxx-x^ in the range : center includegraphicswidth.textwidth#image_path:graph-advanced-doublcapacitor# center abcliste
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Exercise:
A double capacitor consists of two fixed outer plates and two connected inner plates. The inner plates can be moved from left to right thereby changing the effective overlapping plate area of the two parallel plate capacitors with the same plate distance. When the area of the upper capacitor increases the area of the lower one decreases and vice versa see figure. abcliste abc Find an expression for the series capacitance of the two capacitors as a function of the position x. abc Determine the position x_ for the maximum capacitance. abcliste center includegraphicswidth.textwidth#image_path:advanced-doublcapacitor# center
Solution:
abcliste abc With a maximum plate area A and a distance d between the pairs of plates we can write for the two capacitances: C_ varepsilon_fracx Ad C_ varepsilon_frac-x Ad with xin . For the series capacitance we find sscCs leftfracC_+fracC_right^- leftfracdvarepsilon_ x A+fracdvarepsilon_ -x Aright^- varepsilon_fracAdleftfracx+frac-xright^- varepsilon_fracAdleftfrac-x+xx-xright^- varepsilon_fracAd x-x resultvarepsilon_fracAd x-x^ abc In order to find the maximum of the capacitance we can derive the expression from a with respect to the position x: fracmathrmdsscCsmathrmdx varepsilon_fracAd - x At an extremal po xx_ the derivative is equal to zero. This leads to - x_ x_ resultfrac The result can easily be verified with a graph of fxx-x^ in the range : center includegraphicswidth.textwidth#image_path:graph-advanced-doublcapacitor# center abcliste
A double capacitor consists of two fixed outer plates and two connected inner plates. The inner plates can be moved from left to right thereby changing the effective overlapping plate area of the two parallel plate capacitors with the same plate distance. When the area of the upper capacitor increases the area of the lower one decreases and vice versa see figure. abcliste abc Find an expression for the series capacitance of the two capacitors as a function of the position x. abc Determine the position x_ for the maximum capacitance. abcliste center includegraphicswidth.textwidth#image_path:advanced-doublcapacitor# center
Solution:
abcliste abc With a maximum plate area A and a distance d between the pairs of plates we can write for the two capacitances: C_ varepsilon_fracx Ad C_ varepsilon_frac-x Ad with xin . For the series capacitance we find sscCs leftfracC_+fracC_right^- leftfracdvarepsilon_ x A+fracdvarepsilon_ -x Aright^- varepsilon_fracAdleftfracx+frac-xright^- varepsilon_fracAdleftfrac-x+xx-xright^- varepsilon_fracAd x-x resultvarepsilon_fracAd x-x^ abc In order to find the maximum of the capacitance we can derive the expression from a with respect to the position x: fracmathrmdsscCsmathrmdx varepsilon_fracAd - x At an extremal po xx_ the derivative is equal to zero. This leads to - x_ x_ resultfrac The result can easily be verified with a graph of fxx-x^ in the range : center includegraphicswidth.textwidth#image_path:graph-advanced-doublcapacitor# center abcliste
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