Charged Wire (non-uniform)
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
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Exercise:
The charge density in a straight wire of length L is given by lambdax mu absx quad textfor -L/ leq x leq +L/ where mu is given by the length and the charge Q on the wire: mu frac QL^ Derive a formal expression for the electric field at a po on the perpicular bisector to the wire.
Solution:
The solution is similar to that of part a of "Charged Wire" see link with the difference being that the charge density is not constant. Therefore the electric field can be written as Er k_C r _-L/^+L/ fraclambdaxleftr^+x^right^/ textdx k_C r _-L/^+L/ fracmu absxleftr^+x^right^/ textdx The egrand is symmetrical with respect to the erval -L/ +L/ so we only have to calculate on half of the egral: Er k_C r _^+L/ fracmu xleftr^+x^right^/ textdx k_C r mu _^+L/ fracxleftr^+x^right^/ textdx The antiderivative of the egrand is Fx -fracleftr^+x^right^/ so the electric field is Er - k_C r mu left fracleftr^+x^right^/ right_^L/ k_C r mu left fracr - fracsqrtr^+L/^ right k_C fracQL^ left -fracrsqrtr^+L/^ right frac k_C QL^ left-fracsqrt+L/r^ right For a long wire L gg r the second term in the parentheses is much smaller than the first one so the expression can be approximated as Er &approx frac k_C QL^
The charge density in a straight wire of length L is given by lambdax mu absx quad textfor -L/ leq x leq +L/ where mu is given by the length and the charge Q on the wire: mu frac QL^ Derive a formal expression for the electric field at a po on the perpicular bisector to the wire.
Solution:
The solution is similar to that of part a of "Charged Wire" see link with the difference being that the charge density is not constant. Therefore the electric field can be written as Er k_C r _-L/^+L/ fraclambdaxleftr^+x^right^/ textdx k_C r _-L/^+L/ fracmu absxleftr^+x^right^/ textdx The egrand is symmetrical with respect to the erval -L/ +L/ so we only have to calculate on half of the egral: Er k_C r _^+L/ fracmu xleftr^+x^right^/ textdx k_C r mu _^+L/ fracxleftr^+x^right^/ textdx The antiderivative of the egrand is Fx -fracleftr^+x^right^/ so the electric field is Er - k_C r mu left fracleftr^+x^right^/ right_^L/ k_C r mu left fracr - fracsqrtr^+L/^ right k_C fracQL^ left -fracrsqrtr^+L/^ right frac k_C QL^ left-fracsqrt+L/r^ right For a long wire L gg r the second term in the parentheses is much smaller than the first one so the expression can be approximated as Er &approx frac k_C QL^
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Exercise:
The charge density in a straight wire of length L is given by lambdax mu absx quad textfor -L/ leq x leq +L/ where mu is given by the length and the charge Q on the wire: mu frac QL^ Derive a formal expression for the electric field at a po on the perpicular bisector to the wire.
Solution:
The solution is similar to that of part a of "Charged Wire" see link with the difference being that the charge density is not constant. Therefore the electric field can be written as Er k_C r _-L/^+L/ fraclambdaxleftr^+x^right^/ textdx k_C r _-L/^+L/ fracmu absxleftr^+x^right^/ textdx The egrand is symmetrical with respect to the erval -L/ +L/ so we only have to calculate on half of the egral: Er k_C r _^+L/ fracmu xleftr^+x^right^/ textdx k_C r mu _^+L/ fracxleftr^+x^right^/ textdx The antiderivative of the egrand is Fx -fracleftr^+x^right^/ so the electric field is Er - k_C r mu left fracleftr^+x^right^/ right_^L/ k_C r mu left fracr - fracsqrtr^+L/^ right k_C fracQL^ left -fracrsqrtr^+L/^ right frac k_C QL^ left-fracsqrt+L/r^ right For a long wire L gg r the second term in the parentheses is much smaller than the first one so the expression can be approximated as Er &approx frac k_C QL^
The charge density in a straight wire of length L is given by lambdax mu absx quad textfor -L/ leq x leq +L/ where mu is given by the length and the charge Q on the wire: mu frac QL^ Derive a formal expression for the electric field at a po on the perpicular bisector to the wire.
Solution:
The solution is similar to that of part a of "Charged Wire" see link with the difference being that the charge density is not constant. Therefore the electric field can be written as Er k_C r _-L/^+L/ fraclambdaxleftr^+x^right^/ textdx k_C r _-L/^+L/ fracmu absxleftr^+x^right^/ textdx The egrand is symmetrical with respect to the erval -L/ +L/ so we only have to calculate on half of the egral: Er k_C r _^+L/ fracmu xleftr^+x^right^/ textdx k_C r mu _^+L/ fracxleftr^+x^right^/ textdx The antiderivative of the egrand is Fx -fracleftr^+x^right^/ so the electric field is Er - k_C r mu left fracleftr^+x^right^/ right_^L/ k_C r mu left fracr - fracsqrtr^+L/^ right k_C fracQL^ left -fracrsqrtr^+L/^ right frac k_C QL^ left-fracsqrt+L/r^ right For a long wire L gg r the second term in the parentheses is much smaller than the first one so the expression can be approximated as Er &approx frac k_C QL^
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